generalized taylor formulae, computations in real closed valued
TRANSCRIPT
Generalized Taylor formulae,computations in real closed valued fields
and quantifier elimination
Mari-Emi Alonso ∗ Henri Lombardi †
2002
Abstract
We use generalized Taylor formulae in order to give some simple constructions in thereal closure of an ordered valued field. We deduce a new, simple quantifier eliminationalgorithm for real closed valued fields and some theorems about constructible subsets ofreal valuative affine space.
Key words: Valued fields, Real closed fields, Generalized Taylor formulae, Quantifier elimina-tion, Constructive mathematics
MSC 2000: 14P10, 12J10, 12L05, 12Y05, 03F65, 03C10
Introduction
In this work, we consider the real closure of an ordered valued field and search for simple com-putations giving a constructive content to this real closure. We don’t try to give sophisticatedalgorithms which would allow better complexity.
We consider an ordered valued field (K,V,P) with V its valuation ring and P its positivecone. Recall that this means that the following properties hold
V + V ⊆ V, V ×V ⊆ V, ∃x ∈ K \V,∀x, y ∈ K (xy = 1 ⇒ x ∈ V ∨ y ∈ V),
P×P ⊆ P, P + P ⊆ P, ∃x ∈ K \P,∀x, y ∈ K (x+ y = 0 ⇒ x ∈ P ∨ y ∈ P),
∀x, y ∈ K [(x+ y ∈ V, x ∈ P, y ∈ P) ⇒ x ∈ V].
For a, b in K, we write a ≤ b if and only if b− a is in P. We shall use freely in the sequel somewell known features of ordered valued fields: Q ⊆ V, elements of V bounded from below bysome positive rational are units in V, and the non-units in V are the infinitesimal elements ofK.
∗ Universitad Complutense, Madrid, Espana. Partially supported by: PB95/0563-A. M−[email protected]† Laboratoire de Mathematiques, UMR CNRS 6623. Univ. de Franche-Comte, France.
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2
Let S be a subring of V such that K is the fraction field of S. We assume that S is anexplicit ordered ring and that divisibility inside V is testable for two arbitrary elements of S.These are our minimal assumptions of computability. If we want more assumptions in certaincases we shall make them explicit.
We denote the real closure of (K,P) by (Krc,Prc), and we write Vrc for the convex hullof V inside Krc; then Vrc is the unique order-compatible valuation ring extending V. We call(Krc,Vrc,Prc) the real closure of (K,V,P).
In sections 1 and 2 our general purpose is to discuss computational problems in (Krc,Vrc,Prc) under our computability assumptions on (K,V,P).
Each computational problem we shall consider has as input a finite family (ci)i=1,...,n ofparameters in the ring S. We call them the coefficients of our computational problem. Ouralgorithms with the previous minimal computability assumptions work uniformly. This meansthat some computations are made that give polynomials in Z[C1, . . . , Cn], and that all our testsare of the two following types:
Is P (c1, . . . , cn) ≥ 0? Does Q(c1, . . . , cn) divide P (c1, . . . , cn) in V?
We are not interested in how the answers to these tests are found. We may imagine theseanswers given either by some oracles or by some algorithms.
Let us state precisely some other notations. We shall denote the unit group of V by UV,and MV = V \ UV will be the maximal ideal.
We shall denote the residue field V/MV of (K,V) by K, and the value group, K×/UV, byΓK. We use freely the value group’s usual additive notation as well as its usual group-ordering(also denoted by ≤). Recall that ΓKrc is the divisible hull ΓdhK of ΓK. For x ∈ K we write v(x)or vK(x) the valuation of x in ΓK ∪ {+∞}. So
v(0) = +∞, v(xy) = v(x) + v(y), (x ≥ 0, y ≥ 0)⇒ v(x+ y) = min(v(x), v(y)),
and
∀x ∈ K ((v(x) ≥ 0 ⇔ x ∈ V) ∧ (v(x) > 0 ⇔ x ∈MV)).
We write Kac = Krc[√−1] and we denote by Vac the natural valuation ring of Kac extending
Vrc: for a, b ∈ Krc, v(a+ b√−1) = (1/2)v(a2 + b2).
In fact elements of ΓdhK ∪ {+∞} are always defined through elements of S in the followingform. We say that the valuation of some element x belonging to Kac is well determined if weknow integers m and n, elements c1, ..., cn in S, and two elements F and G of Z[C1, ..., Cn], suchthat, setting f = F (c1, . . . , cn) (f 6= 0) and g = G(c1, . . . , cn), there exists a unit u in Vac with:
fxm = ug
(in particular, v(0) is well determined).We read the previous formula as:
mvKac(x) = vK(g)− vK(f),
or more simply as:
mv(x) = v(g)− v(f).
For x, y ∈ Kac, we shall use the notation x � y for v(x) ≤ v(y) (i.e., y = x = 0 ∨ (x 6=0 ∧ y/x ∈ Vac)).
3
Example. Let us explain the computations that are necessary to compare 3v(x1) + 2v(x2) to7v(x3) when the valuations are given by
f1xm11 = u1g1, f2x
m22 = u2g2, f3x
m33 = u3g3 (g1, g2, g3 6= 0).
We consider the LCM m = m1n1 = m2n2 = m3n3 of m1,m2,m3. We have
fn11 xm1 = un1
1 gn11 , f
n22 xm2 = un2
2 gn22 , f
n33 xm3 = un3
3 gn33 .
So 3v(x1) + 2v(x2) ≤ 7v(x3) iff g3n11 g2n2
2 f 7n33 � f 3n1
1 f 2n22 g7n3
3 .
The reader can easily verify that computations we shall run in the value group are alwaysmeaningful under our computability assumptions on the ring S.
In the same way, elements of the residue field will in general be defined from elements of V.So computations inside the residue field are given by computations inside S.
We now give an outline of the paper.In section 1 we give some basic tools used in the rest of the paper. First we recall the
Newton Polygon Algorithm and the Generalized Tschirnhaus Transformation. Then we insiston Generalized Taylor formulae, which are formulae giving P (x) on a Thom interval as a sumof terms all having the same sign. This feature allows us to give a good description for v(P (x))with the crucial Theorem 1.3.6. This allows us to give a nice description for “constructible”subsets of the real line in the context of real closed valued fields (cf. Theorem 1.4.4).
In section 2 we settle three basic computational problems in the real closure of an orderedvalued field. We solve the first problem by a simple trick (subsection 2.3). The consequence isthat when we know how to compute in a given ordered valued field, we know how to computein its real closure. This can be seen as a not too difficult extension of basic algorithms in realclosed fields. Solving the second problem is possible by using our first algorithm, but we preferto develop another algorithm, similar to the Cohen-Hormander algorithm for ordered fields. Weget in this way nice uniform results describing precisely some generalizations of the completetableau of signs in the real closed case (Theorems 2.4.5 and 2.5.2).
In section 3 we give parametrized versions of previous algorithms (Theorems 3.1.1 and3.1.3), and we apply these results to quantifier elimination in real closed valued fields. We con-sider the first order theory of real closed valued fields based on the language of ordered fields(0, 1,+,−,×,=,≤) to which we add the predicate x � y. So, all constants and variables repre-sent elements in K (this corresponds to our previously explained computability assumptions).We get the following theorem.
Theorem 3.2.1 Let Φ(a, x) be a quantifier free formula in the first order theory of realclosed valued fields. We view the ai’s as parameters and the xj’s as variables.
Then one can give a quantifier free formula Ψ(a) such that the two formulae ∃x Φ(a, x)and Ψ(a) are equivalent in the formal theory. (The terms appearing in the formulae Φ and Ψare Z–polynomials in the parameters, and, in the case of Φ, also in the variables.)
We think we have given here a rather simple proof of this fundamental, well known result(see e.g., [2]).
We also get the following abstract form of the previous theorem.
Theorem 3.3.2 Let us denote the real-valuative spectrum of a commutative ring A bySpervA. Then the canonical mapping from SpervA[X] to SpervA transforms any constructiblesubset into a constructible subset.
In section 4 we apply the parametrized algorithms in order to study constructible subsets(in the meaning of real closed valued fields). First, we get the analogue of the Tarski-Seidenbergprinciple.
4 1 BASIC MATERIAL
Theorem 4.1.1 Let (K,V,P) be an ordered valued subfield of a real closed valued field(R,VR,PR). Let π be the canonical projection from Rn+r onto Rn. Let S ⊆ Rn+r be any(≤,�)–constructible set defined over (K,V,P). Assume that the sign test and the divisibilitytest are explicit inside the ring generated by the coefficients of the polynomials that appear in thedefinition of S. Then a description of the projection π(S) ⊆ Rn can be computed in a uniformway by an algorithm that uses only rational computations, sign tests and divisibility tests.
In particular, the complexity of a description of π(S) is explicitly bounded in terms of thecomplexity of a description of S.
Finally we construct a kind of stratification for (≤,�)–constructible sets, that we call strat-ification a la Cohen Hormander because it is a further development of the same notion for semi-algebraic sets (cf. [1] chapter 9), and we finish the paper with the following cell-decompositiontheorem (for a precise definition of Q-semilinear functions see definition 2.4.4).
Theorem 4.2.5 (Cell decomposition theorem) Let (K,V,P) be an ordered valued subfieldof a real closed valued field (R,VR,PR). Let g1, . . . , gs be nonzero polynomials in K[x1, . . . , xn].Consider a linear change of variables together with a family (fi,j)i=1,...,n;j=1,...,`i that give a strat-ification for (g1, . . . , gs). Assume that this stratification is constructed a la Cohen-Hormander.Consider any k-dimensional stratum Cε corresponding to this stratification. Then there is aNash isomorphism
h : (R+)k −→ Cε, (t1, . . . , tk) 7−→ h(t1, . . . , tk)
with the following property.If S is any (≤,�)–constructible subset described from g1, . . . , gs, then S ∩ Cε is a finite
union of cells h(Li), where each Li can be defined as{(t1, . . . , tk) ∈ (R+)k :
∧`
a`(τ) = α` ∧∧m
bm(τ) > βm
},
where τ = (τ1, . . . , τk) = (v(t1), . . . , v(tk), the a`’s and bm’s are Z-linear forms w.r.t. τ , andα`, βm ∈ ΓdhK .
Moreover, each τi is a Q-semilinear function in some v(Fj(x1, . . . , xn))’s (with Fj explicitlycomputable in K[x1, . . . , xn]).
1 Basic material
1.1 The Newton Polygon
Here we recall the well known Newton Polygon algorithm.A multiset is a set with (nonnegative) multiplicities, or equivalently a list defined up to
permutation. E.g., the roots of a polynomial P (X) repeated according to multiplicities form amultiset in the algebraic closure of the base field. We shall use the notation [x1, . . . , xd] for themultiset corresponding to the list (x1, . . . , xd). The cardinality of a multiset is the length of acorresponding list, i.e., the sum of multiplicities occurring in the multiset.
The Newton polygon of a polynomial P (X) =∑
i=0,...,d piXi ∈ K[X] (where pd 6= 0) is
obtained from the list of pairs in N× (ΓK ∪ {+∞})
((0, v(p0)), (1, v(p1)), . . . , (d, v(pd))).
The Newton polygon is “the bottom convex hull” of this list. It can be formally defined asthe extracted list ((0, v(p0)), . . . , (d, v(pd))) verifying: two pairs (i, v(pi)) and (j, v(pj)) are twoconsecutive vertices of the Newton polygon iff:
1.2 Generalized Tschirnhaus transformation 5
if 0 ≤ k < i then (v(pj)− v(pi))/(j − i) > (v(pi)− v(pk)/(i− k))if i < k < j then (v(pk)− v(pi))/(k − i) ≥ (v(pj)− v(pi))/(j − i)if j < k ≤ d then (v(pk)− v(pj))/(k − j) > (v(pj)− v(pi))/(j − i)
It is easily shown that if (i, v(pi)) and (j, v(pj)) are two consecutive vertices in the Newtonpolygon of the polynomial P , then the zeroes of P in Kac whose valuation in ΓdhK equals(v(pi)− v(pj))/(j − i) form a multiset with cardinality j − i.
Computational problem 0 (Multiset of valuations of roots of polynomials)Input: Let P ∈ K[X] be a polynomial over a valued field (K,V).Output: The multiset [v(x1), . . . , v(xn)] where [x1, . . . , xn] is the multiset of roots of P in Kac.
Newton Polygon algorithmThe number n∞ of roots equal to 0 (i.e., with infinite valuation) is read on P . Let P0 := P/Xn∞ .Compute the Newton polygon of P0, compute the slopes and output the answer.
1.2 Generalized Tschirnhaus transformation
We recall here the well known (generalized) Tschirnhaus transformation, which we will usefreely in our computations.
Let K be a field, (Pj)j=1,...,m be a family of monic polynomials in K[X], and
Pj(X) = (X − xj,1)× · · · × (X − xj,dj)
their decompositions in Kac. Let Q(Y1, . . . , Ym) be a polynomial in K[Y1, . . . , Ym]. Then thepolynomial
TQ(Z) = (Z −Q(x1,1, . . . , xm,1))× · · · × (Z −Q(x1,d1 , . . . , xm,dm))
is the characteristic polynomial of AQ where AQ is the matrix of the multiplication byQ(y1, . . . , ym) inside the d-dimensional K-algebra
K[y1, . . . , ym] := K[Y1, . . . , Ym]/ 〈P1(Y1), . . . , Pm(Ym)〉
(d = d1 · · · dm, and yi is the class of Yi modulo 〈P1, . . . , Pm〉).Now let R ∈ K[Y1, . . . , Ym] with R(x) 6= 0 for all m-tuples x = (x1,r1 , . . . , xm,rm). So AR is
an invertible matrix. Let F = Q/R, then the polynomial
TF (Z) = (Z − F (x1,1, . . . , xm,1))× · · · × (Z − F (x1,d1 , . . . , xm,dm))
is the characteristic polynomial of AQ(AR)−1.
1.3 Generalized Taylor Formulas
Using the usual Taylor formula for computing valuations in ΓdhK .For P ∈ K[X] we denote P [k] = P (k)/k!, where P (k) is the k-th derivative of P . Let t = x−a,
and assume deg(P ) = d, the usual Taylor formula at the point a is
P (x) = P (a) + P [1](a)t+ P [2](a)t2 + · · ·+ P [d−1](a)td−1 + P [d]td.
Now assume that P [d] > 0. Let a0 be the greatest real root of the product PP [1] · · ·P [d−1]. Ifa ≥ a0 we see that all P [k](a) are ≥ 0 and we get the following expression for the valuationv(P (x)) when x > a
v(P (x)) = min(ν0, ν1 + τ, ν2 + 2τ, . . . , νd + dτ)
6 1 BASIC MATERIAL
where τ = v(t) and νj = v(P [j](a)) (some νj’s may be infinite). So, w.r.t. the variable τ thevaluation of P (x) in ΓdhK is piecewise linear and increasing. Note that τ decreases from +∞ to−∞ when t increases from 0 to +∞.
In the following paragraphs, we see that generalized Taylor formulae allow us to give asimilar description of the valuation v(P (x)) when x is inside a Thom interval.
What are generalized Taylor formulae?A fundamental example of algebraic evidence for a sign is given by generalized Taylor formu-
lae, which make explicit some consequences of Thom’s lemma in terms of algebraic identities.Thom’s lemma implies that the set of points where a real polynomial and its successive
derivatives have fixed signs is an interval. An easy proof, by induction on the degree of thepolynomial, is based on the mean value theorem. We can translate this geometric fact underthe form of algebraic identities called Generalized Taylor Formulas (GTF for short).
Let us see an example where deg(P ) ≤ 4.
Example 1.3.1 Consider the general polynomial of degree 4
P (X) = c0X4 + c1X
3 + c2X2 + c3X + c4,
consider the following system of sign conditions for the polynomial P and its successive deriva-tives with respect to the variable X:
H(X) : P (X) > 0, P [1](X) < 0, P [2](X) < 0, P [3](X) < 0, P [4] > 0.
Consider also the system of sign conditions obtained by relaxing all the inequalities, except oneof them, e.g., the last one:
H ′(X) : P (X) ≥ 0, P [1](X) ≤ 0, P [2](X) ≤ 0, P [3](X) ≤ 0, P [4] > 0.
Thom’s lemma implies that:
[H ′(a), H ′(b), a < x < b ] =⇒ H(x).
Put e1 = x− a, e2 = b− x. Consider the following algebraic identity in Z[c0, . . . , c4, a, b, x]
P (x) = P (b)− e2 P [1](a)− (2e1e2 + e22)P [2](a)
−(3e12e2 + 3e1e2
2 + e23)P [3](b)
+(8e13e2 + 12e1
2e22 + 12e1e2
3 + 3e24)P [4].
This gives clearly an evidence that, when P ∈ K[X] where K is an ordered field,
[H ′(a), H ′(b), a < x < b ] =⇒ P (x) > 0.
One can find more information about mixed and generalized Taylor formulae in [6, 10, 11].The important thing is that for any fixed degree, and any combination of signs for P and itsderivatives (which are assumed to be fixed on the interval), there exists a corresponding GTF.We state a general result giving the existence of GTF’s.
Proposition 1.3.2 (see [10]) Let P be a polynomial of degree d in K[X] and a, b, x threevariables. Let e1 = x − a, e2 = b − x. Let ε = (ε1, . . . , εd) be any sequence in {−1,+1}. Letε0 = 1. Then there exists an algebraic identity
P (x) = P (a0) +d−1∑k=1
εkHk,ε(e1, e2)P[k](ak) + εdHd,ε(e1, e2)P
[d]
1.3 Generalized Taylor Formulas 7
where each polynomial Hk,ε is homogeneous of degree k with nonnegative integer coefficients,ak = a if εkεk+1 = 1, and ak = b if εkεk+1 = −1.
Moreover, if ε1 = 1, then e1 divides all the Hk,ε’s, and the coefficient of ek1 in Hk,ε is nonzero.In a similar way if ε1 = −1, then e2 divides all the Hk,ε’s, and the coefficient of ek2 in Hk,ε isnonzero.
Remark 1.3.3 Let P be a polynomial of degree d in K[X] and let a < b ∈ Krc be such thatP [k](a)P [k](b) ≥ 0 for k = 0, . . . , d. This gives a system of signs (σ0, σ1, . . . , σd) (σi = ±1)(σk is the sign of P [k](x) on the open interval ]a, b[ ). Let εi = σ0σi, ε = (ε1, . . . , εd). Thenthe corresponding GTF gives an algebraic certificate for the fact that sign(P (x)) = σ0 whena < x < b.
We now give four GTF’s in degree 3, those beginning by P (a) + e1P[1] · · ·. Each formula is
given also with e1 in factor in the second part.
P (x) = P (a) + e1P[1](a) + e21P
[2](a) + e31P[3]
= P (a) + e1(P [1](a) + e1P
[2](a) + e21P[3])
= P (a) + e1P[1](a) + e21P
[2](b)− (2e31 + e21e2)P[3]
= P (a) + e1(P [1](a) + e1P
[2](b)− (2e21 + e1e2)P[3])
= P (a) + e1P[1](b)− (e21 + 2e1e2)P
[2](a)− (2e31 + 6e21e2 + 3e1e22)P
[3]
= P (a) + e1(P [1](b)− (e1 + 2e2)P
[2](a)− (2e21 + 6e1e2 + 3e22)P[3])
= P (a) + e1P[1](b)− (e21 + 2e1e2)P
[2](b) + (e31 + 3e21e2 + 3e1e22)P
[3]
= P (a) + e1(P [1](b)− (e1 + 2e2)P
[2](b) + (e21 + 3e1e2 + 3e22)P[3]).
There are also four other GTF’s beginning by P (b) − e2.P [1] · · ·. They can be obtained fromthe first ones by swapping a and b, and replacing e1 and e2 by −e2 and −e1
P (x) = P (b)− e2P [1](b) + e22P[2](b)− e32P [3]
= P (b)− e2(P [1](b)− e2P [2](b) + e22P
[3])
= P (b)− e2P [1](b) + e22P[2](a) + (2e32 + e22e1)P
[3]
= P (b)− e2(P [1](b)− e2P [2](a)− (2e22 + e2e1)P
[3])
= P (b)− e2P [1](a)− (e22 + 2e2e1)P[2](b) + (2e32 + 6e22e1 + 3e2e
21)P
[3]
= P (b)− e2(P [1](a) + (e2 + 2e1)P
[2](b)− (2e22 + 6e2e1 + 3e21)P[3])
= P (b)− e2P [1](a)− (e22 + 2e2e1)P[2](a) + (e32 − 3e22e1 + 3e2e
21)P
[3]
= P (b)− e2(P [1](a) + (e2 + 2e1)P
[2](a) + (e22 + 3e2e1 + 3e21)P[3]).
Using generalized Taylor formulae for computing the variations of the valuationv(P (x)).
Now let us see in the case of an ordered valued field how these formulae can be used in orderto describe the variations of v(P (x)) when x is on the real line Krc.
Example 1.3.4 Let a, b ∈ Krc and assume that the signs of the derivatives of a polynomial Pof degree 4 are the same in a and b, as in Example 1.3.1. If x ∈ [a, b] let x = a + t1(b − a),
8 1 BASIC MATERIAL
e = b − a, e1 = t1e, e2 = t2e (so t2 = 1 − t1), δ = v(e), τ1 = v(t1), τ2 = v(t2), ν0 = v(P (b)),ν1 = v(P [1](a)), ν2 = v(P [2](a)), ν3 = v(P [3](b)), ν4 = v(P [4]). We rewrite the GTF as
P (x) = P (b)− e t2 P [1](a)− e2(2t1t2 + t22)P [2](a)
−e3(3t12t2 + 3t1t22 + t2
3)P [3](b)+e4(8t1
3t2 + 12t12t2
2 + 12t1t23 + 3t2
4)P [4].
In the above GTF, since all terms of the sum are ≥ 0, the valuation of the sum is the minimumof valuations of the terms, so we get:
(1) If t1 and t2 are units, then τ1 = τ2 = 0 and v(P (x)) is constant equal to
v(P (x)) = min(ν0, ν1 + δ, ν2 + 2δ, ν3 + 3δ, ν4 + 4δ).
(2) If t1 is infinitely close to 0, then τ1 > 0 (decreasing as t1 increases), τ2 = 0, and v(P (x))is a priori increasing “piecewise linearly w.r.t. τ1”, but in our case constant
v(P (x)) = min(ν0, ν1 + δ, ν2 + 2δ, ν3 + 3δ, ν4 + 4δ).
(3) If t1 is infinitely close to 1, then τ1 = 0, τ2 > 0 (increasing as t1 increases), and v(P (x))is increasing “piecewise linearly w.r.t. τ2”
v(P (x)) = min(ν0, ν1 + δ + τ2, ν2 + 2δ + τ2, ν3 + 3δ + τ2, ν4 + 4δ + τ2).
In fact here we see that this formula is true in the three cases and that only two slopes(w.r.t. the variable τ2) can appear since
v(P (x)) = min(ν0, min(ν1 + δ, ν2 + 2δ, ν3 + 3δ, ν4 + 4δ) + τ2).
Example 1.3.5 In a similar way let us see what is given by the second GTF in degree 3
P (x) = P (a) + e1 P[1](a) + e21 P
[2](b)−(2e31 + e21e2
)P [3].
We assume P (a) ≥ 0, P [1](a) ≥ 0, P [2](b) ≥ 0, P [3] < 0, x = a + t1(b − a), e = b − a,e1 = t1e, e2 = t2e (t2 = 1− t1), δ = v(e), τ1 = v(t1), τ2 = v(t2), ν0 = v(P (a)), ν1 = v(P [1](a)),ν2 = v(P [2](b)), ν3 = v(P [3]), and we get
P (x) = P (a) + e t1 P[1](a) + e2t21 P
[2](b)− e3(t31 + t21t2
)P [3].
(1) If t1 and t2 are units, then τ1 = τ2 = 0 and v(P (x)) is constant equal to
v(P (x)) = min(ν0, ν1 + δ, ν2 + 2δ, ν3 + 3δ).
(2) If t1 is infinitely close to 1, then τ1 = 0, τ2 > 0 (increasing as t1 increases), and v(P (x))is increasing “piecewise linearly w.r.t. τ2”, but in our case constant
v(P (x)) = min(ν0, ν1 + δ, ν2 + 2δ, ν3 + 3δ).
(3) If t1 is infinitely close to 0, then τ1 > 0 (decreasing as t1 increases), τ2 = 0, and v(P (x))is increasing “piecewise linearly w.r.t. τ1”,
v(P (x)) = min(ν0, ν1 + δ + τ1, ν2 + 2δ + 2τ1, ν3 + 3δ + 2τ1).
In fact here we see that this formula is true in the three cases and that only three slopes(w.r.t. the variable τ1) can appear since
v(P (x)) = min(ν0, (ν1 + δ) + τ1, min(ν2 + 2δ, ν3 + 3δ) + 2τ1).
1.4 Constructible subsets of the real line 9
What we have seen on our two Examples 1.3.4 and 1.3.5 is a general result, that we imme-diately get as a corollary of Proposition 1.3.2.
Theorem 1.3.6 Let P be a polynomial of degree d in K[X] and a < b ∈ Krc such thatP [k](a)P [k](b) ≥ 0 for k = 0, . . . , d. Let (σ0, σ1, . . . , σd) be the signs of P, P [1], . . . , P [d] in theinterval ]a, b[ (σi = ±1). Let εi = σ0σi, ε = (ε1, . . . , εd). Let us consider the correspondingGTF as in Proposition 1.3.2, and let us follow the notation there. Let νk = v(P [k](ak)) fork = 0, . . . , d. Recall that ak = a if εkεk+1 = 1, and ak = b if εkεk+1 = −1. Note also that νkmay be infinite if k < d. If x ∈ [a, b] let x = a + t1(b − a), e = b − a, t2 = 1 − t1, δ = v(e),τ1 = v(t1), τ2 = v(t2).
Then for x ∈ [a, b] the valuation v(P (x)) is monotonic w.r.t. t1 and more precisely can bedescribed in the following way.
(a) – If ε1 = 1 we can extract from the GTF integers k2, . . . , kd such that 1 ≤ kj ≤ j and
v(P (x)) = min(ν0, ν1 + δ + τ1, ν2 + 2δ + k2τ1, . . . , νd + dδ + kdτ1).
– If ε1 = −1 we can extract from the GTF integers k2, . . . , kd such that 1 ≤ kj ≤ j and
v(P (x)) = min(ν0, ν1 + δ + τ2, ν2 + 2δ + k2τ2, . . . , νd + dδ + kdτ2).
(b) So in any case the valuation v(P (x)) is
– either constant (if v(P (a)) = v(P (b))),
– or increasing piecewise linearly w.r.t. τ1 = v(x−ab−a ) (if v(P (a)) > v(P (b))),
– or increasing piecewise linearly w.r.t. τ2 = v( b−xb−a), (if v(P (a)) < v(P (b))).
(c) Introducing
τ = τ1 − τ2 = v
(t1
1− t1
)we also get: τ1 = max(τ, 0) = τ+, τ2 = max(−τ, 0) = τ−, and the value v(P (x)) ismonotone and piecewise linear w.r.t. τ . More precisely, we can extract from the GTFintegers k2, . . . , kd such that 1 ≤ kj ≤ j and
v(P (x)) = min(ν0, ν1 + δ + τ ′, ν2 + 2δ + k2τ′, . . . , νd + dδ + kdτ
′)
where τ ′ = max(ε1τ, 0).
1.4 Constructible subsets of the real line
We introduce here the notion of (≤,�)–constructible sets in the real valuative affine space. Thisnotion corresponds to sets that are definable in the language of ordered valued fields. Thesesets are analogous to Zariski-constructible sets in algebraic geometry and to semi-algebraic setsin real algebraic geometry.
Definition 1.4.1 Let (K,V,P) be an ordered valued field, and consider a finite family(xj)j=1,...,m of elements of Krc. Let us call a valued sign condition (a vsc fort short) for thefamily any condition of the following type
∧j∈J
sign(xj) = σj ∧∧`∈L
sign
∑j∈J, σj 6=0
`jv(xj)
= σ′`
10 1 BASIC MATERIAL
where J ⊆ {1, . . . ,m}, ` ∈ L (L is a finite subset of Z{j : j∈J, σj 6=0}) and σj, σ′` ∈ {−1, 0, 1}.
Let N be a positive integer. We call an N -complete system of valued sign conditionson the family (xj)j=1,...,m a system of vsc’s that gives all the signs sign(xj) and all the signs
sign(∑
xj 6=0 `jv(xj))
for all ` ∈ {−N, . . . , 0, . . . , N}{j : 1≤j≤m,xj 6=0}.
An alternative definition could use sign(∑
j∈J, `jv(xj))
even when v(xj) =∞ for some j’s.
But there should be no natural way to give a sign to an expression containing ∞−∞.
Definition 1.4.2 Let (K,V,P) be an ordered valued subfield of a real closed valued field (R,VR,PR), and consider a finite family (Pj)j=1,...,m of polynomials in K[X1, . . . , Xn].
• The subset of Rn made of the x = (x1, . . . , xn) such that the Pj(x)’s verify some givensystem of vsc’s is called a basic (≤,�)–constructible set defined over (K,V,P).
• A (general) (≤,�)–constructible set defined over (K,V,P) is any boolean combinationS of basic (≤,�)–constructible sets defined over (K,V,P). If (Pj)j=1,...,m is a family ofpolynomials such that any basic component of S is defined as in the first item, we say thatS is described from (Pj)j=1,...,m.
• Let S ⊆ Rn be a (≤,�)–constructible set. A map f : S → Rp is called a (≤,�)–cons-tructible map if its graph is a (≤,�)–constructible subset of Rn+p.
Let us recall that the order topology and the valued topology are identical in a real closedvalued field.
Notation 1.4.3 Let (K,V,P) be an ordered valued subfield of a real closed valued field (R,VR,PR). We shall use the following notations for some convex open (≤,�)–constructible subsetsof the real line. They are basic (≤,�)–constructible sets defined over (Krc,Vrc,Prc).
I+(a, α) = {x ∈ R : x = a+ t, 0 < t, v(t) = α}with a ∈ Krc, α ∈ ΓdhK .
I−(a, α) = {x ∈ R : x = a− t, 0 < t, v(t) = α}with a ∈ Krc, α ∈ ΓdhK .
I+(a, α, β) = {x ∈ R : x = a+ t, 0 < t, α < v(t) < β}with a ∈ Krc, α < β in ΓdhK ∪ {±∞}.
I−(a, α, β) = {x ∈ R : x = a− t, 0 < t, α < v(t) < β}with a ∈ Krc, α < β in ΓdhK ∪ {±∞}.
J+(a, b, α) = {x ∈ R : x = a+ t(b− a), 0 < t, v(t) = α}with a < b ∈ Krc, 0 < α ∈ ΓdhK .
J−(a, b, α) = {x ∈ R : x = b− t(b− a), 0 < t, v(t) = α}with a < b ∈ Krc, 0 < α ∈ ΓdhK .
J+(a, b, α, β) = {x ∈ R : x = a+ t(b− a), 0 < t, α < v(t) < β}with a < b ∈ Krc, 0 ≤ α < β ∈ ΓdhK ∪ {+∞}.
J−(a, b, α, β) = {x ∈ R : x = b− t(b− a), 0 < t, α < v(t) < β}with a < b ∈ Krc, 0 ≤ α < β ∈ ΓdhK ∪ {+∞}.
J(a, b) = {x ∈ R : x = a+ t(b− a), 0 < t < 1, v(t) = v(1− t) = 0}with a < b ∈ Krc.
These subsets will be called (<,�)–intervals defined over (K,V,P).
Some remarks.
1.4 Constructible subsets of the real line 11
• The subsets of R given in definition 1.4.2 are a priori basic (≤,�)–constructible setsdefined over (Krc,Vrc,Prc). But they are also general (≤,�)–constructible sets definedover (K,V,P): this is a consequence of Remark 3.1.2 (2).
• In J+(a, b, α), J−(a, b, α), J+(a, b, α, β) and J−(a, b, α, β) we have 0 < t < 1 (in fact t <any positive rational number) since t > 0 and v(t) > 0.
• Except when β =∞, any (<,�)–interval is closed.
• We have
]a,∞[ = I+(a,−∞,∞),]a, b[ = J+(a, b, 0,∞) ∪ J(a, b) ∪ J−(a, b, 0,∞),
I+(a, α, γ) = I+(a, α, β) ∪ I+(a, β) ∪ I+(a, β, γ) if α < β < γ,
and similar results with I−, J+ and J−.
• When t > 0, α < β ∈ ΓdhK ∪ {+∞}, c > 0 ∈ K and v(c) = α + β we have the followingequivalences
α < v(t) < β ⇐⇒ α < min(v(t), v(c/t)) ⇐⇒α < v(t+ c/t) ⇐⇒ α + v(t) < v(t2 + c).
• Concerning J(a, b) we have
J(a, b) = {x ∈ R : x = a+ t(b− a), 0 < t(1− t), v(t(1− t)) = 0} .
• All J’s could be considered as particular cases of I’s, e.g., J+(a, b, α, β) = I+(a, α′, β′) withα′ = α + v(b− a) and β′ = β + v(b− a).
• We could introduce
J(a, b, α) = {x ∈ R : x = a+ t(b− a), 0 < t < 1, v(t/(1− t)) = α}with a < b ∈ Krc, α ∈ ΓdhK ,
J(a, b, α, β) = {x ∈ R : x = a+ t(b− a), 0 < t < 1, α < v(t/(1− t)) < β}with a < b ∈ Krc, α < β in ΓdhK ∪ {±∞}.
We should have J+(a, b, α) = J(a, b, α), J−(a, b, α) = J(a, b,−α), J+(a, b, α, β) =J(a, b, α, β), J−(a, b, α, β) = J(a, b,−β,−α) and J(a, b) = J(a, b, 0).
An easy corollary of Theorem 1.3.6 is the following description of (≤,�)–constructiblesubsets of the real line.
Theorem 1.4.4 Let (K,V,P) be an ordered valued subfield of a real closed valued field (R,VR,PR). Any (≤,�)–constructible set of R defined over (K,V,P) is a finite disjoint unionof points in Krc and of (<,�)–intervals defined over (K,V,P) as in Notations 1.4.3.
W e give a sketch of the proof on an example. Assume that the (≤,�)–constructible set Sis defined from vsc’s on 3 polynomials P1, P2, P3 of degrees 5, introduce all real roots of thesepolynomials and of all their derivatives. Consider two consecutive roots a, b. We want tounderstand what S ∩ ]a, b[ is.
First let us see what S ∩ J+(a, b, 0,∞) looks like. We know that each sign(Pj(x)) is constanton ]a, b[ . Concerning the valuations v(Pj(x)), we know from Examples 1.3.4 and 1.3.5 and
12 2 COMPUTING IN THE REAL CLOSURE OF AN ORDERED VALUED FIELD
Theorem 1.3.6 that they are piecewise linear functions of τ1 = v(x − a)/v(b − a), e.g., of thefollowing forms
v(P1(x)) = min(µ0, µ1 + τ1, µ2 + 2τ1),v(P2(x)) = min(η0, η1 + τ1, η3 + 3τ1),v(P3(x)) = min(λ0, λ1 + τ1, λ2 + 2τ1, λ4 + 4τ1).
Note that τ1 varies on ]0,+∞[ . These piecewise linear functions have polygonal graphsinside (ΓdhK∩ ]0,+∞[ ) × ΓdhK . It is possible to compute the vertices of these three polygonalgraphs. E.g., if λ4 < λ1 < λ0 and 3λ2 > 2λ1 + λ4 we have two vertices on the polygonal graphof v(P3) at the points with coordinates
τ1,1 = β1 = (λ1 − λ4)/3, v(P3(x)) = λ1 + β1 = λ4 + 4β1,τ1,2 = β2 = λ0 − λ1, v(P3(x)) = λ0 = λ1 + β2.
All these vertices give a finite number of valuations for τ1: α1 < · · · < αn. Let α0 = 0,αn+1 =∞. On each J+(a, b, αi, αi+1) (0 ≤ i ≤ n) and on each J+(a, b, αi) (1 ≤ i ≤ n), we knowthat each v(Pj(x)) (1 ≤ j ≤ 3) is a fixed “affine function” of τ1. So, the same is true for anylinear combination
`1v(P1(x)) + `2v(P2(x)) + `3v(P3(x)),
and we can compute the valuation τ1 for which such an expression changes sign.So the intersection S ∩ J+(a, b, 0,∞) is a finite disjoint union of J+(a, b, α, β) and J+(a, b, α)
subsets.In a similar way S ∩ J(a, b) is either empty or equal to J(a, b), and S ∩ J−(a, b, 0,∞) is a
finite disjoint union of J−(a, b, α, β) and J−(a, b, α) subsets.Finally the intersection of S with the final (resp. initial) open interval is computed in a
similar way as a finite union of I+ (resp. I−) intervals. 2
2 Computing in the real closure of an ordered valued
field
2.1 Codes a la Thom and valuations in the value group
The real closure Krc of an ordered field (K,P) is unique up to unique (K,P)–isomorphism.This fact allows us to give an explicit construction of the real closure Krc (this is “well-known”from Tarski or even from Sturm and Sylvester, for a fully constructive proof see [7]).
E.g., it is possible to describe any element x of Krc by a so-called code a la Thom (see [3, 4]):
Definition 2.1.1 A pair (P, σ) where P ∈ K[X] is a monic polynomial of degree d and σ =(σ1, . . . , σd−1) ∈ {1,−1}d−1 codes the root x of P in Krc when one has
P (x) = 0 and σi · P (i)(x) ≥ 0 for i = 1, . . . , d− 1.
The pair (P, σ) is called a code a la Thom (over K) for x.
There are algorithms that use only the algebraic structure of (K,P) and give the codesa la Thom corresponding to the roots of P in Krc. It is possible to make explicit algebraiccomputations and sign’s tests for such elements that are coded a la Thom. See e.g., [3, 4] orProposition 2.4.2.
On the other hand, the Newton polygon algorithm allows us to determine the valuationv(x) for any x in the algebraic closure of K. How can we match these algorithms?
2.2 Three basic computational problems in the real closure of an ordered valued field 13
2.2 Three basic computational problems in the real closure of anordered valued field
Consider an ordered valued field (K,V,P). Since its real closure (with valuation) is determinedup to unique (K,V,P)-isomorphism, the following computational problems makes sense:
Computational Problem 1Let (K,V,P) be an ordered valued field.Input: A code a la Thom (P, σ) over K for an element x of Krc.Output: The valuation v(x) of x in ΓdhK ∪ {+∞}. More precisely, compute some a ∈ K and apositive integer n such that n× v(x) = v(a).
Remark 2.2.1 Assume that the leading coefficient of P ∈ V[X] is a unit. The real zeroes ofP are in Vrc. Let us denote by x the residue in Krc of the zero x and by P the residue in K[X]of the polynomial P . Then it is clear that (P , σ) is a code a la Thom over K for x since theresidual field Krc can be identified with the real closure K
rcof K.
More generally, we can ask for algorithms solving general existential problems.
Computational Problem 2Let (K,V,P) be an ordered valued field, and consider a finite family of polynomials, (Fj)j=1,...,m
in K[X]. Let (xh)h=1,...,p be the ordered family of the zeroes of the (Fj)’s in Krc. Recall that thenumber p and all the signs sign(Fj(x)), for x equal to some xh or inside some correspondingopen interval, can be determined by computations in the ordered field (K,P).Input: The family (Fj)j=1,...,m.Output: All the valuations v(Fj(xh)) (h = 1, . . . , p) and v(xh+1 − xh) (h = 1, . . . , p − 1) inΓdhK ∪ {+∞}.
Computational Problem 3Let (K,V,P) be an ordered valued field.Input: A finite family (Fj)j=1,...,m in K[X]. A finite family (`k)k=1,...,r of elements of Zm.Output: All occurring systems of valued sign conditions of the following type for the family(Fj(x))j=1,...,m when x ∈ Krc:(sign(Fj(x)))j=1,...,m,
sign
∑j∈{1,...,m}, Fj(x)6=0
`k,j v(Fj(x))
k=1,...,r
.
Remark 2.2.2 Assume that the family is stable under derivation. From Theorem 1.3.6 (seee.g., the proof of Theorem 1.4.4) it is clear that Computational Problem 3 can be solved by usingthe solution of Computational Problem 2. In fact we can describe in a finite way all occurringlists
( (sign(Fj(x)))j=1,...,m, (v(Fj(x)))j=1,...,m )
when x ∈ Krc: for x on any (<,�)–interval I used in the proof of Theorem 1.4.4 we havev(Fj(x)) = µI,j +mI,jv(t) where t is either (x− xh)/(xh+1− xh), or (xh+1− x)/(xh+1− xh), orx1 − x or x− xp.
14 2 COMPUTING IN THE REAL CLOSURE OF AN ORDERED VALUED FIELD
2.3 Solving the first problem
Algorithm RCVF1 solving Problem 1. Recall that (P, σ) is a code a la Thom fora root x of P ∈ K[X]. We can assume w.l.o.g. that P (0) 6= 0, x > 0 (else replace P byP (−X)) and that P is monic. Let (xi)i=1,...,d be the roots of P in Kac. Using the NewtonPolygon algorithm, we compute the multiset [ v(xi) | i = 1, . . . , n ]. So we can express the setof valuations v(xi) as (v(cj)/nj)j=1,...,r for some r-tuple (cj, nj)j=1,...,r with cj > 0 in K, nj ∈ Nand v(cj)/nj < v(cj+1)/nj+1 for j = 1, . . . , r − 1.Consider the LCM n of denominators nj and “replace each xi by zi = xni ”: i.e., compute
Q(X) =∏
i(X − zi) and compute a code a la Thom (Q, σ′) for z = xn. Let bj = cn/nj
j . Thenv(bj) = (n/nj)v(cj) for j = 1, . . . , r and
v(b1) < · · · < v(br).
So we have alsob1 > · · · > br > 0.
By rational computations in (K,P) we can settle one of the three following inequalities in Krc
z ≥ b1,bj ≥ z ≥ bj+1 with some j ∈ {1, . . . , r − 1},
br ≥ z > 0.
In the first case we conclude that v(z) = v(b1). In the last case v(z) = v(br). In the remainingcase we know that
v(bj) ≤ v(z) ≤ v(bj+1) so v(z) = v(bj) or v(z) = v(bj+1).
We have to find the exact valuation. Consider c ∈ P verifying
0 < v(c) ≤ min
(v
(bjbj−1
), v
(bj+1
bj
))if j > 1
0 < v(c) = v
(b2b1
)if j = 1
(if j > 1, c can be chosen as bj/bj−1 or bj+1/bj). Next consider the linear fractional change ofvariable
y 7−→ ϕ(y) =y
1 + cy2
We have— If v(y) ≥ 0 then v(ϕ(y)) = v(y).— If v(y) ≤ −v(c) then, letting y′ = 1/y we get
v(y′) ≥ v(c) > 0, ϕ(y) =y′
c+ y′2and v(ϕ(y)) = v(y′)− v(c) ≥ 0.
So the monic polynomial
R(Y ) =∏i
(Y − ϕ
(zibj
))has coefficients in V. Moreover v(z/bj) ≥ 0, so ϕ(z/bj) is a unit iff v(bj) = v(z) sincev(ϕ(z/bj)) = v(z/bj).
2.4 Solving the second problem 15
We can compute a code a la Thom (R, σ′′) for ϕ(z/bj). This gives a code a la Thom (R, σ′′)
for ϕ(z/bj) (i.e., ϕ(z/bj) considered as an element of Krc
). Finally we test whether this codeis verified by 0 (which is a root of R). In case of negative answer then v(z) = v(bj). Otherwisev(z) = v(bj+1).
Remarks 2.3.11) In a more explicit view, we should ask for computing two nonnegative elements a and b
of K and an integer n such that a ≤ |x|n ≤ b and v(a) = v(b).2) Clearly algorithm RCVF1 allows us to run sure computations inside (Krc,Vrc,Prc) when
we know how to compute inside (K,V,P).
2.4 Solving the second problem
First we recall the Cohen-Hormander algorithm for ordered fields (see e.g., [1] chapter 1).
Definition 2.4.1 Let (K,P) be an ordered field and (Fj) a finite family of univariate polyno-mials in K[X]. A complete tableau of signs for the family (Fj) is the following discrete dataT :
• The ordered list (xk)k=1,...,r of all the roots of all the Fj’s in Krc.
• The signs (∈ {−1, 0,+1}) of all the Fj’s at all the xk’s.
• The signs of all the Fj’s in each interval ] − ∞, x1[ , ]xk, xk+1[ (1 ≤ k ≤ r − 1) and]xr,+∞[ .
We call an xk a point of the tableau T . Similarly an interval ] − ∞, x1[ or ]xk, xk+1[ or]xr,+∞[ is called an interval of the tableau T .
In this tableau xk is merely a name for the corresponding root, it may be coded by thenumber k or in another way.
Proposition 2.4.2 (Cohen-Hormander’s algorithm for computing the complete tableau ofsigns for a finite family of univariate polynomials) Let (K,P) be an ordered subfield of areal closed field (R,PR). Let L = (F1, . . . , Fk) be a list of polynomials in K[Y ]. Let L′ bethe family of polynomials generated by the elements of L and by the operations P 7→ P ′ and(P,Q) 7→ Rem(P,Q) for deg(P ) ≥ deg(Q) ≥ 1. Then L′ is finite and one can compute thecomplete tableau of signs for L′ in terms of the following data:
• the degree of each polynomial in the family L′,
• the diagrams of operations P 7→ P ′ and (P,Q) 7→ Rem(P,Q),
• the signs of constants ∈ L′.
L et us remark that in this algorithm the zero polynomial can appear in L′ as a remainderRem(P,Q) where deg(P ) ≥ deg(Q) ≥ 1. The degree of the zero polynomial is −1.
The list L′ is finite: one makes systematically the operation “derivation of every previouslyobtained polynomial” and “remainders of all previously obtained couple of polynomials”, andone gets a finite family at the end since degrees are decreasing.
Let us number the polynomials in L′ with an order compatible with the order on thedegrees. Let L′m be the subfamily of L′ made of polynomials numbered from 1 to m. This
16 2 COMPUTING IN THE REAL CLOSURE OF AN ORDERED VALUED FIELD
family is obviously stable under the operations “derivation” and “remainder by a division”which decrease strictly the degrees. Denote lastly by Tm the corresponding complete tableauof signs.
We are going to prove, by induction on m, that the complete tableau of signs of the polyno-mials in the family L′m can be obtained by using only the authorized informations. As long aspolynomials are of degree 0, this is clear. Suppose it is true up to m. Let P be the polynomialof number m+ 1 in L′. On each interval of Tm, the polynomial P is strictly monotonic. Everypoint a of Tm is either +∞, or −∞, or a root of a certain polynomial Q with number ≤ m,and in this case, if R = Rem(P,Q), we have P (a) = R(a). The sign of P (a) is hence known inevery case from the authorized informations. This allows us to know on which open intervals ofTm the polynomial P has a root in R. Let x be such a root of P on one of these open intervalsI = ]a, b[ . If Q is a polynomial of number ≤ m in P , its sign on the interval I is known. Thismeans we know its sign at the point x, and on intervals ]a, x[ and ]x, b[. With respect to P ,its signs on ]a, x[ and on ]x, b[ are also known since P is strictly monotonic on the interval.The complete tableau of signs for L′m+1 is thus known from the authorized informations andthe complete tableau of signs for L′m. 2
In this algorithm we remark that each zero of the tableau is obtained with a Thom’s encod-ing.
An extension of previous algorithm will solve Problem 2. First we give a valued version forthe complete tableau of signs.
Definition 2.4.3 Let (K,V,P) be an ordered valued field and (Fj)j∈J a finite family of uni-variate polynomials in K[X]. A complete tableau of vsc’s for the family (Fj) is the followingdata T :
• The ordered list (xk)k=1,...,r of all the roots of all the Fj’s in Krc.
• The complete tableau of signs for the family (Fj)j∈J .
• All the valuations v(xk+1 − xk) (k = 1, . . . , r − 1).
• All the valuations v(Fj(xk)) (j ∈ J, k = 1, . . . , r).
Algorithm RCVF2 solving Problem 2. A first possibility is to use algorithm RCVF1.We think that it is interesting to indicate another possibility which goes in the same spiritas the Cohen-Hormander algorithm for ordered fields. This gives us also simple proofs fortheorems in sections 3 and 4. Call (Pj) the list L′ in Proposition 2.4.2. Call (xm,k)k=1,...,rm
the ordered list of all roots of L′m = (Pj)j=1,...,m. We replace in the proof of Proposition 2.4.2the complete tableau of signs Tm of L′ by Sm = Tm ∪ Vm where Vm collects the valuationsv(Pj(xm,k)) (j ∈ {1, . . . ,m}, k ∈ {1, . . . , rm}) and v(xm,k+1 − xm,k) (k ∈ {1, . . . , rm − 1}.)
Suppose we have done the job up to m. Let P = Pm+1 be the polynomial of index m + 1in L′. The tableau Tm+1 is computed as in Proposition 2.4.2. It remains to compute missinginformations in Vm+1.
At every root a = xm,k of a polynomial Q = P` with index ` ≤ m, if R = Rem(P,Q), wehave P (a) = R(a) and R is in L′m, so the valuation v(P (a)) is known from Vm.
Let x = a+ t1(b− a) be a root of P on an open interval I = ]xm,k, xm,k+1[ = ]a, b[ of Tm. Inorder to compute all the v(Pj(x))j=1,...,m it is sufficient to compute v(t1) = τ1 and v(t2) = τ2(t2 = 1− t1): Theorem 1.3.6 says us how to get the valuations v(Pj(x))j=1,...,m from Vm, τ1 andτ2.
2.4 Solving the second problem 17
In order to compute τ1 = v(t1) we use a GTF that expresses P (x) = P (a+ t1(b− a)) = 0 as
P (a) + t1 ·
(d∑j=1
εj · ej ·Gj,ε(t1, t2) · P [j](aj)
)(aj = a or b)
where e = b− a, t1 ·Gj,ε(t1, t2) = Hj,ε(t1, t2) and
sign(εjP[j](aj)) = sign(−P (a)) (1 ≤ j ≤ d).
Moreover, the valuations v(P (a) = ν, v(P [j](aj)) = νj and δ = v(b − a) are known. From theproperties of Hj,ε, we know that Gj,ε(t1, t2) is a unit if τ1 = 0, so its valuation in ΓdhK ∪ {+∞}depends only on τ1. So we get
v(P (a)) = ν = min(ν1 + δ + τ1, ν2 + 2δ + k2τ1, . . . , νd + dδ + kdτ1)
(τ1 ≥ 0, and some νk’s may be infinite). The right hand side is an increasing piecewise linearfunction of τ1 so we have a unique and explicit solution τ1. With µi = νi + iδ we precisely get
τ1 = max
(ν − µ1,
ν − µ2
k2
, . . . ,ν − µdkd
).
Finally τ2 is computed analogously and we can fill up Vm+1.Remark also that if x is on the last interval ]xm,rm ,+∞[ = ]a,+∞[ of Tm, we can compute
v(x− a) in a similar way by using the usual Taylor formula.
Definition 2.4.4 In an additive divisible ordered group G we consider terms built from vari-ables αj by Q-linear combinations and by using the operations min and max. We call such aterm a Q-semilinear term. The function defined by such a term is called a Q-semilinear functionof the αj’s.
We get the following theorem, similar to Proposition 2.4.2.
Theorem 2.4.5 (An algorithm a la Cohen-Hormander for computing the complete tableau ofvsc’s for a finite family of univariate polynomials) Let (K,V,P) be an ordered valued subfieldof a real closed valued field (R,VR,PR). Let L = (F1, . . . , Fk) be a list of polynomials in K[Y ].Let L′ be the (finite) family of polynomials generated by the elements of L and by the operationsP 7→ P ′ and (P,Q) 7→ Rem(P,Q) for deg(P ) ≥ deg(Q) ≥ 1. Call (cj) the list of constants∈ L′.
Then one can compute the complete tableau of vsc’s for L′ in terms of the following data:
• the degree of each polynomial in the family,
• the diagrams of operations P 7→ P ′ and (P,Q) 7→ Rem(P,Q) in L′,
• the signs sign(cj),
• the valuations v(cj).
Moreover, all the valuations v(xk+1 − xk) and all the valuations v(Pj(xk)) are given as fixedQ-semilinear functions of the v(cj)’s: each such Q-semilinear function is a fixed Q-semilinearterm (in the “variables” v(cj)’s) that depends only on the complete tableau of signs of L′.
T his theorem is an extension of Proposition 2.4.2. The proof is similar. In fact we get allresults by a close inspection of Algorithm RCVF2. 2
18 2 COMPUTING IN THE REAL CLOSURE OF AN ORDERED VALUED FIELD
2.5 Solving the third problem
Algorithm RCVF3 solving Problem 3. We run Algorithm RCVF2 and we applyTheorem 1.3.6: see Remark 2.2.2.
Definition 2.5.1 Let (Fj)j∈J be a finite family of univariate polynomials in K[X] (where (K,V,P) is an ordered valued field). We assume the family to be stable under derivation. Let Mbe a positive integer.
An M -complete tableau of vsc’s for the family (Fj) is the following discrete data T :
• The ordered list (xk)k=1,...,r of all the roots of all the Fj’s in Krc.
• For each k = 1, . . . , r, the M-complete system of vsc’s (see Definition 1.4.1) for the family(Fj(xk))j∈J .
• For each k = 1, . . . , r − 1
– The M-complete system of vsc’s for the family (Fj(x))j∈J for x ∈ J(xk, xk+1).
– A partition of J+(xk, xk+1, 0,∞) as a finite union of 2nk + 1 (<,�)–intervals⋃i=0,nk
J+(xk, xk+1, αk,i, αk,i+1) ∪⋃
i=1,nk
J+(xk, xk+1, αk,i),
(where αk,0 = 0 and αk,nk+1 = ∞) and for each (<,�)–interval A of this partition,the M-complete system of vsc’s for the family (Fj(x))j∈J which is the same one forany x ∈ A.
– A similar data concerning J−(xk, xk+1, 0,∞).
• Similar data concerning I−(x1,−∞,∞) and I+(xr,−∞,∞).
In this tableau the αk,i’s (0 < αk,1 < · · · < αk,nk<∞) are purely formal and nk is the only
relevant information concerning αk,1, . . . , αk,nk.
We now state a result that precises the output of Algorithm RCVF3.
Theorem 2.5.2 (An algorithm a la Cohen-Hormander for computing an M -complete tableauof vsc’s for a finite family of univariate polynomials)Let (K,V,P) be an ordered valued subfield of a real closed valued field (R,VR,PR). LetM be a positive integer. Let L = (F1, . . . , Fk) be a list of polynomials in K[Y ]. Let L′ bethe family of polynomials generated by the elements of L and by the operations P 7→ P ′ and(P,Q) 7→ Rem(P,Q) for deg(P ) ≥ deg(Q) ≥ 1. Call (cj) the list of constants ∈ L′.
Then one can compute the M-complete tableau of vsc’s for L′ in terms of the following data:
• the degree of each polynomial in the family,
• the diagrams of operations P 7→ P ′ and (P,Q) 7→ Rem(P,Q) in L′,
• the N-complete system of vsc’s for the family (cj),
where N is an integer depending only on M and on the list of degrees in L.
19
3 Quantifier elimination algorithms
3.1 Parametrized computations
Algorithms RCVF2 and RCVF3 are uniform: they can be run when coefficients in the initialdata are polynomials in other variables which are called parameters (instead of being in thebase field).
A case by case discussion appears, and the straight-line algorithm is replaced by a branchingone.
We describe this situation as a parametrized algorithm dealing with parametrized univariatepolynomials.
Theorem 3.1.1 (parametrized version of Theorem 2.4.5) Let (K,V,P) be an ordered valuedsubfield of a real closed valued field (R,VR,PR). Let L = (F1, . . . , Fk) be a list of parametrizedunivariate polynomials of degrees d1, . . . , dk in some variable X. Let us run the algorithmRCVF2 and let us open two branches in the computation any time we have to know if a givenelement is zero or nonzero when computing a remainder. Moreover, replace remainders bypseudoremainders in order to avoid denominators.
Consider the family (cj) of all “constants” in all L′’s that appear at the leaves of the tree(these constants are K–polynomials in the parameters).
Finally consider that the computed valuations v(xk+1 − xk) and v(Pj(xk)) at any leave ofthe tree are given as Q-semilinear functions of the “variables” v(cj)’s.
Then this global parametrized algorithm is finite and therefore gives a finite number of pos-sibilities for its output: the complete tableau of vsc’s for L.
More precisely when the signs of the “constants” cj’s are known, the complete tableau of signsis known and all the valuations v(xk+1 − xk) and v(Pj(xk)) are given as explicit Q-semilinearfunctions in the “variables” v(cj)’s.
T he proof of Proposition 2.4.2 (Cohen-Hormander algorithm) works as well in theparametrized case. In each branch so created, the proof of Theorem 2.4.5 works as well. 2
Remarks 3.1.2
1) An important case is obtained when all coefficients of the Fi’s are independent param-eters and (K,V,P) = (Q,Q,Q≥0). This “generic case” gives the complete description of allsituations occurring with a fixed number of polynomials of known degrees.
2) Another interesting particular case is the following one, with only one parameter subjectto certain constraints. We start with a list of polynomials L = (F1, . . . , Fk) in K[Y ]n, we getan extended list L′ and the complete tableau of signs. Let a and b be two consecutive rootsin this tableau. Now we want to make computations with an element x of the interval ]a, b[ .Consider x as a parameter verifying some sign constraints, namely the Thom’s sign conditionsthat define ]a, b[ . We add the polynomial Y − x to L and we run the parametrized versionof RCVF2. Only one root is added: x. The new polynomials appearing are only “constants”of the form Q(x) (where Q is in L′). The process go on only trough one branch. We get thefollowing result: the valuations v(x − a) and v(b − x) are given as Q-semilinear functions ofsome v(Q(x))’s. From this we also get a similar result concerning v(x − xj) where xj is anyroot in the tableau. Naturally, there is also a parametrized version for this result.
Similarly we have a parametrized version of Theorem 2.5.2.
20 3 QUANTIFIER ELIMINATION ALGORITHMS
Theorem 3.1.3 (parametrized version of Theorem 2.5.2)Let L = (F1, . . . , Fk) be a list of parametrized univariate polynomials of degrees d1, . . . , dk insome variable X. Let M be a positive integer. Let us run the algorithm RCVF2 and let usopen two branches in the computation any time we have to know if a given element is zero ornonzero when computing a remainder. Moreover, replace remainders by pseudoremainders inorder to avoid denominators. Let us call (cj) the family of all “constants” in all L′’s that appearat the leaves of the tree (these constants are K–polynomials in the parameters).
Finally when applying Theorem 1.3.6 in order to get the output of RCVF3 from the oneof RCVF2, we open three branches any time we have to know the sign of some Z-linearcombination of v(cj)’s.
Then this global parametrized algorithm is finite and therefore gives a finite number of pos-sibilities for its output: the M-complete tableau of vsc’s for L.
Moreover, these outputs depend on the following data:
• the signs of the “constants” cj’s,
• the sign test inside a finite subset of the subgroup generated by the v(cj)’s; which areexactly divisibility tests between monomials in the cj’s).
Remark 3.1.4 Since the computation in the previous theorem is purely formal, certain systemsof conditions corresponding to the data given by the two last items may be impossible. If we wantto know what are these impossible systems, we have to use the quantifier elimination algorithmgiven in Theorem 3.2.2. Nevertheless, one can verify that there is no circular argument.
3.2 Quantifier elimination
We now give some corollaries of previous computations for quantifier elimination. We recallthat these results are well known, see e.g., [2].
We consider the first order theory of real closed valued fields based on the language ofordered fields (0, 1,+,−,×,=,≤) to which we add the predicate x � y. So, all constants andvariables represent elements in K (this corresponds to our previously explained computabilityassumptions).
Here is a corollary of Theorem 3.1.3.
Theorem 3.2.1 Let Φ(a, x) be a quantifier free formula in the first order theory of real closedvalued fields. We view the ai’s as parameters and the xj’s as variables. Then one can give aquantifier free formula Ψ(a) such that the two formulae ∃x Φ(a, x) and Ψ(a) are equivalentin the formal theory. (The terms appearing in the formulae Φ and Ψ are Z–polynomials in theparameters, and, in the case of Φ, also in the variables.)
U se recursively Theorem 3.1.3 and eliminate the xj’s one after the other. 2
We also get the following corollary.
Theorem 3.2.2 Let (K,V,P) be an ordered valued subfield of a real closed valued field (R,VR,PR). Assume that the sign test and the divisibility test are explicit inside (K,V,P). Thenthere is a uniform quantifier elimination algorithm for the first order theory of real closed valuedfields extending (K,V,P).
3.3 An abstract form of quantifier elimination 21
3.3 An abstract form of quantifier elimination
An abstract form of Theorem 3.1.3 is the following theorem, that was given the first time byM.J. De la Puente in [9].
First, we need some definitions of the abstract objects.
Definition 3.3.1 Let us denote the real-valuative spectrum of a commutative ring A bySpervA: an element of SpervA is given by a ring homomorphism ϕ from A to a real closedvalued field K, and two such homomorphisms ϕ, ϕ′ define the same element of SpervA iffthere exists an isomorphism of ordered valued fields ψ : R → R′ such that ψ ◦ ϕ = ϕ′,where R and R′ are the real closed valued fields generated by ϕ(A) and ϕ′(A). Alterna-tively, an element of SpervA is given by a prime ideal Q of A and a structure of orderedvalued field upon the fraction field of A/Q. A constructible subset of SpervA is by definition aboolean combination of elementary constructible subsets Ux := {ϕ ∈ SpervA : ϕ(x) > 0} andVx,y := {ϕ ∈ SpervA : ϕ(x) � ϕ(y)}, where x, y ∈ A.
Theorem 3.3.2 The canonical mapping from SpervA[X] to SpervA transforms any (≤,�)–constructible subset into a (≤,�)–constructible subset.
A (≤,�)–constructible subset in Sperv(B) is a finite union of basic (≤,�)–constructible sub-sets, that are defined as
{ϕ ∈ Sperv(B) :∧i ϕ(ai) = 0 ∧
∧j ϕ(bj) > 0 ∧
∧k v(ϕ(ck)) = v(ϕ(dk)) ∧
∧` v(ϕ(e`)) > v(ϕ(f`))
}where conjunctions are finite and all elements are in B. Searching the canonical image of abasic constructible subset S of SpervA[X] (defined by elements ai, bj, ck, dk, e`, f` in A[X]) insideSpervA, is the same thing that analyzing the conditions on the coefficients of the polynomialsai, bj, ck, dk, e`, f` allowing the existence of an x where the defining conditions of S are verified.So Theorem 3.1.3 gives the answer. 2
Another consequence of Theorem 3.1.3 is a relativized version of Theorem 3.3.2. Thisgeneralization is obtained by giving some constraints on the ring homomorphism ϕ from A toa real closed valued field K. We give e.g., a subring B of A, an ideal M of B, a multiplicativemonoid S in A and a semi ring P in A (P + P ⊆ P, P × P ⊆ P ). We want to allow onlyhomomorphisms φ (from A or A[X] to a real closed valued field) verifying that φ(B) is in thevaluation ring, φ(M) is in the maximal ideal, elements of φ(S) are nonzero and elements of φ(P )are nonnegative. If we write C the constraints (B,M, S, P ) and if we write Sperv(A,C) thepart of SpervA satisfying the constraints, we get: the canonical mapping from Sperv(A[X], C)to Sperv(A,C) transforms any (≤,�)–constructible subset in a (≤,�)–constructible subset.
In [9] the relativized version is settled with one constraint B.
4 Constructible subsets in the real valuative affine space
4.1 Tarski-Seidenberg-Chevalley
We now give a geometric form for Theorems 3.1.3 and 3.2.2.
Theorem 4.1.1 Let (K,V,P) be an ordered valued subfield of a real closed valued field (R,VR,PR). Let π the canonical projection from Rn+r onto Rn. Let S ⊆ Rn+r be any (≤,�)–constructible set defined over (K,V,P). Assume that the sign test and the divisibility test
22 4 CONSTRUCTIBLE SUBSETS IN THE REAL VALUATIVE AFFINE SPACE
are explicit inside the ring generated by the coefficients of the polynomials that appear in thedefinition of S. Then a description of the projection π(S) ⊆ Rn can be computed in a uniformway by an algorithm that uses only rational computations, sign tests and divisibility tests.
In particular, the complexity of a description of π(S) is explicitly bounded in terms of thecomplexity of a description of S.
Here rational computations mean computations in the ring generated by the coefficientsof the polynomials occurring in the description of S. A description of S is a quantifier freeformula in disjunctive normal form describing S. The complexity of such a description of S canbe defined as a 5-tuple (n, d, k, `,m) where n is the number of variables, d is the maximum ofthe degrees, k is the number of polynomials, ` is the number of ∨ and m is the bound for thenumbers of ∧ inside a disjunct.
Corollary 4.1.2 Let (K,V,P) be an ordered valued subfield of a real closed valued field (R,VR,PR). Let S ⊆ Rn be a (≤,�)–constructible set and let f : S → Rp be a (≤,�)–construc-tible map.
• The interior and the adherence of S inside Rn for the order topology are (≤,�)–cons-tructible sets.
• f(S) ⊆ Rp is a (≤,�)–constructible set.
• Let T be a (≤,�)–constructible set containing f(S) and let g : T → Rq be a (≤,�)–cons-tructible map. Then g ◦ f is a (≤,�)–constructible map.
• Let T ′ ⊆ Rp be a (≤,�)–constructible set. Then f−1(T ′) ⊆ Rn is a (≤,�)–constructibleset.
4.2 Stratifications and applications
We think that the results of this section could allow to get most of the results obtained byFrank Mausz in his Doctoral dissertation [8] with a different approach.
Lojaziewicz stratification a la Cohen-HormanderWe recall here a result about stratifying families ([1] chapter 9).
Definition and notation 4.2.1 Consider a general monic polynomial of degree d as a pointof Rd. Let σ = (σ1, . . . , σd) ∈ {−1,+1}d. Let
Uσ =
{P ∈ Rd : ∃x ∈ R
(P (x) = 0 ∧
d∧i=1
sign(P (i)(x)) = σi
)}.
It is easily seen that Uσ is a connected open semialgebraic subset of Rd (see e.g., [5]) and that
Uσ =
{P ∈ Rd : ∃x ∈ R
(P (x) = 0 ∧
d∧i=1
sign(P (i)(x)) ∈ {σi, 0}
)}.
For P ∈ Uσ we call ρσ(P ) the zero which is coded a la Thom by (P, σ). Then P 7→ ρσ(P ) isNash on Uσ and admits a continuous semialgebraic extension on Uσ, that we note also by ρσ.Such a function will be called a Thom’s root function, or simply a root function.
More generally, if ϕ : Rk−1 → Rd is a polynomial function, we can consider ρσ ◦ ϕ asdefined over ϕ−1(Uσ). We also call such a function a root function. This function is Nash
4.2 Stratifications and applications 23
over ϕ−1(Uσ). If f(x1, . . . , xk) = ϕ(x1, . . . , xk−1)(xk) is the corresponding monic polynomial ink variables, we denote ρσ ◦ ϕ by ρσ(f).
Finally if a polynomial g ∈ K[x1, . . . , xk] = K[x1, . . . , xk−1][xk] has a leading coefficientw.r.t. xk which is a nonzero element c of K, we say that g is quasi monic in xk, and we letρσ(g) = ρσ(g/c).
For more details about root functions see [5].
Theorem 4.2.2 ([1] chap. 9) Let (K,P) be an ordered subfield of a real closed field (R,PR).Let g1, . . . , gs be nonzero polynomials in K[x1, . . . , xn]. After a suitable linear change of variablesthere exists a family of polynomials
(fi,j)i=1,...,n;j=1,...,`i
with the following properties (we will continue denoting the new variables by xi).
(1) First we have
– (g1, . . . , gs) ⊆ (fn,j)j=1,...,`n
– Each fk,j is a nonzero polynomial in K[x1, . . . , xk] which is quasimonic in xk.
– For each index k the family (fk,j)j=1,...,`k is stable under derivation w.r.t. xk (exclud-ing the zero derivative).
(2) Let us denote Ik = {(i, j) : i = 1, . . . , k; j = 1, . . . , `i}. Call Ck the family of nonemptysemialgebraic subsets of Rk that can be defined as some
Cε =
(ξ1, . . . , ξk) ∈ Rk ;∧
(i,j)∈Ik
sign(fi,j(ξ1, . . . , ξk)) = εi,j
6= ∅(where ε = (εi,j)(i,j)∈Ik is any family in {−1, 0,+1}). It is clear that the Cε’s in Ck give apartition of Rk. We have
(a) The canonical projection πk(Cε) of any element Cε ∈ Ck on Rk−1 is an element ofCk−1: it is obtained as Cε′ where ε′ is the restriction of the family ε to Ik−1.
(b) The adherence Cε of Cε (recall we assume Cε 6= ∅) is a union of elements of Ck, itis obtained by relaxing strict inequalities in the definition of Cε.
(c) If in the definition of Cε ∈ Ck there is one equality fk,i(ξ1, . . . , ξk) = 0 then Cε isthe graph of a root function ρσ(fk,j) (here fk,j is seen as a polynomial in xk, it is
equal to fk,i or to some f(`)k,i and σ is extracted from ε) which is Nash over πk(Cε).
Moreover, ρσ(fk,i) is defined over πk(Cε) and the graph of this root function is Cε.
(d) Call πn,k the canonical projection Rn → Rk. Let E be a k dimensional semialgebraicsubset of Rn defined from the polynomials g1, . . . , gs. Then for any Cε ∈ Cn which iscontained in E, πn,k maps homeomorphically Cε on its image.
Definition 4.2.3 Such a change of variables together with such a family (fi,j) will be called astratification for (g1, . . . , gs) and for any semialgebraic subset of Rn defined from this family.The family (fi,j)i=1,...,n;j=1,...,`i will be called a stratifying family for the initial family (g1, . . . , gs).The semialgebraic subsets Cε are called the strata of the stratification.
24 4 CONSTRUCTIBLE SUBSETS IN THE REAL VALUATIVE AFFINE SPACE
We shall precisely consider the following way of constructing a stratifying family, a la Cohen-Hormander (it is the one suggested in [1].) First we make a linear change of variables in orderto make g1, . . . , gs quasi monic in the new variable xn. We add all the derivatives of each giw.r.t. xn. This gives us the family (fn,j)j=1,...,`n .
We apply Cohen-Hormander’s algorithm to this family and we call h1, . . . , h` the “constants”given by this algorithm (these constants are polynomials in (x1, . . . , xn−1)).
We make a new linear change of variables on (x1, . . . , xn−1) in order to make h1, . . . , h`quasi monic in the new variable xn−1. We make the same linear change of variables inside(fn,j)j=1,...,`n : this family remains quasimonic in xn and stable under derivation w.r.t. xn, andh1, . . . , h` remain the “constants” given by the Cohen-Hormander’s algorithm when applied tothis family.
We add all the derivatives of each hi w.r.t. xn−1. This gives us the family (fn−1,j)j=1,...,`n−1 .And so on.
With this kind of stratifying family, we can apply recursively Theorem 3.1.3. So we get aprecise description of the variation of the valuations v(fk,j(x1, . . . , xk)) when (x1, . . . , xk) ∈ Cεfor any k and any Cε ∈ Ck. Let us see an example.
Example 4.2.4 Assume n = 3. Consider a cell C ∈ C3. Assume that C ′′ = π3,1(C) is aninterval ]a, b[ , that C ′ = π3,2(C) is the graph of a root function h1 = ρσ(f2,1) defined on [a, b],and that C is the part of C ′ ×R between two root functions h2 = ρσ′(f3,1) and h3 = ρσ′′(f3,2),so
C = {(x, y, z) : a < x < b, y = h1(x), h2(x, y) < z < h3(x, y)}= {(x, y, z) : a < x < b, y = h1(x), h′2(x) < z < h′3(x)} .
We consider for (x, y, z) ∈ C, the parameters t = (x − a)/(b − x), τ = v(t), t′ = (z −h′2(x))/(h′3(x)− z) and τ ′ = v(t′). We get:
• The map h : (t, t′) 7→ (x, y, z) ∈ C is a Nash isomorphism from (R+)2 onto C.
• For any fk,j in the stratifying family v(fk,j(x, y, z)) = ϕk,j(τ, τ′) is a Q-semilinear function
of τ, τ ′ (here we use recursively Theorem 3.1.3).
• So, if we look at C ∩ S where S is any (≤,�)–constructible subset described from thefk,j’s, we find that C ∩ S is a finite union of sets h(Li) where each Li is defined as{
(t, t′) ∈ (R+)2 :∧`
a`(τ, τ′) = α` ∧
∧m
bm(τ, τ ′) > βm
}
where a`’s and bm’s are Z-linear forms and α`, βm ∈ ΓdhK .
• Now we should like to have some rational expression of τ and τ ′ that uses only polynomialsin (x, y, z). This is possible in the following way, as in Remark 3.1.2. Consider thatthe formal variables are X, Y, Z and that x, y, z are three parameters. Add to the listgi the three polynomials X − x, Y − y, Z − z and reconstruct the stratification, usingthe information that (x, y, z) is in the semialgebraic set C. You get that τ and τ ′ arefixed Q-semilinear functions in the v(cj)’s and in some v(Fj(x, y, z))’s: the cj’s are theold constants, and the Fj(x, y, z) are the new “constants” that are constructed by thealgorithm (Fj(x, y, z) ∈ K[x, y, z]).
The following “cell decomposition theorem” is merely the generalization of what we haveseen on this example. It is obtained by applying Theorem 1.3.6 to a stratification a la Cohen-Hormander. The last assertion is obtained as in Remark 3.1.2.
REFERENCES 25
Theorem 4.2.5 (Cell decomposition theorem) Let (K,V,P) be an ordered valued subfield ofa real closed valued field (R,VR,PR). Let g1, . . . , gs be nonzero polynomials in K[x1, . . . , xn].Consider a linear change of variables together with a family (fi,j)i=1,...,n;j=1,...,`i that give a strati-fication for (g1, . . . , gs). Assume that this stratification is constructed a la Cohen-Hormander, asexplained above (after Definition 4.2.3). Consider any k-dimensional stratum Cε correspondingto this stratification (see Theorem 4.2.2). Then there is a Nash isomorphism
h : (R+)k −→ Cε, (t1, . . . , tk) 7−→ h(t1, . . . , tk)
with the following property.If S is any (≤,�)–constructible subset described from g1, . . . , gs, then S ∩ Cε is a finite
union of cells h(Li), where each Li can be defined as{(t1, . . . , tk) ∈ (R+)k :
∧`
a`(τ) = α` ∧∧m
bm(τ) > βm
}
where τ = (τ1, . . . , τk) = (v(t1), . . . , v(tk), the a`’s and bm’s are Z-linear forms w.r.t. τ , andα`, βm ∈ ΓdhK .
Moreover, each τi is a Q-semilinear function in some v(Fj(x1, . . . , xn))’s (with Fj’s explicitlycomputable elements of K[x1, . . . , xn]).
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26 CONTENTS
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[11] Warou H. Formules de Taylor Generalisees et applications. Preprint Universite de Niamey(1999). 6
Contents
Introduction 1
1 Basic material 41.1 The Newton Polygon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Generalized Tschirnhaus transformation . . . . . . . . . . . . . . . . . . . . . . 51.3 Generalized Taylor Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Constructible subsets of the real line . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Computing in the real closure of an ordered valued field 122.1 Codes a la Thom and valuations in the value group . . . . . . . . . . . . . . . . 122.2 Three basic computational problems in the real closure of an ordered valued field 132.3 Solving the first problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 Solving the second problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.5 Solving the third problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 Quantifier elimination algorithms 193.1 Parametrized computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Quantifier elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3 An abstract form of quantifier elimination . . . . . . . . . . . . . . . . . . . . . 21
4 Constructible subsets in the real valuative affine space 214.1 Tarski-Seidenberg-Chevalley . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.2 Stratifications and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 22