intersection of quadrics and optimality issues dupont, lazard father and son, petitjean (loria and...
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Intersection of quadrics and Intersection of quadrics and optimality issuesoptimality issues
Dupont, Lazard father and son, Petitjean (LORIA and LIP6)
JGA’03
Main result (SoCG’03)Main result (SoCG’03)
Input: 2 implicit quadrics, rational coefficients
near-optimaloptimal
1 extra rational point on conic
rational point on a degree-8 surface
Output: exact parametric representation of intersection Worse-case optimal and always near-optimal
in the degree of the field extension on which the coefficients are defined
Outline of improvementsOutline of improvements
Starting point: Levin’s pencil method
“Simple” (but substantial) improvements: projective setting Gauss reduction of quadratic forms nice parameterizations of projective quadrics
More “involved” improvements: quadric through a rational point avoid explicit reduction ad hoc algorithm for every case
Worth it ??
Levin (1976, 1979)Levin (1976, 1979)
Input: and Find a (simple) ruled quadric in Find frame where is canonical
P transformation matrix, X parameterization Solve degree 2 equation Plug result in X and compute PX
Nested (ruled quadric)Nested (eigenvectors)another (parameterization)
TS QQ
01 PXSPX T
SQ TQRQ
RQ
nestingdepth 5 !!
““Improved projective Levin”Improved projective Levin”
Recall that: Euclidean projective projective quadrics are characterized by
inertia ruled quadrics are those of inertia (3,1) Gauss reduction is rational optimal parameterization of (2,2):
““Improved projective Levin”: Improved projective Levin”: algoalgo
Input: and projective quadrics
Find a quadric of inertia (2,2) in
Find frame where is canonical (Gauss) P transformation matrix, X parameterization
Solve degree 2 equation
Plug result in X and compute PX
SQ TQ
RQ TS QQ RQ
0PXSPX TT
““Improved projective Improved projective Levin”Levin”
Result is in:
Height of the coefficients: d: input quadrics height of output is
ddd 1021768
)()( )( 413 uuPuP
More improvments (generic More improvments (generic case)case)
Features: find a quadric of inertia (2,2) in the pencil
going through a rational point
do not use Gauss directly (but use the rational point found above)
optimize transformations
Near-optimal Near-optimal algorithmalgorithm
Result is in
Height: d: input quadrics, p: rational point height of output is
Observed heights: p = 0 (uncorrelated) random data [-1000,+1000]: 20d (gcd) real data: 5d
52p42d)128(3829 pdpd
)()( )( 413 uuPuP
Near-optimal output: exampleNear-optimal output: example Example coming from real
application (modeling of a tea pot with SGDLsoft)
You could even be lucky!You could even be lucky!
Kill two square roots with one stone…
Can we go further? Can we go further?
In the smooth quartic case, asserting whether the is needed amounts to
finding a rational quadric of the pencil going througha rational point and whose determinant is a square
1. Find a rational point on hyperelliptic curve:
2. Find a rational point on associated quadric:
Rational points on curves and Rational points on curves and surfacessurfaces
ratpoints by Michael Stoll (Univ. of Bremen)
Generically, no solution (computational number theory) confirmed by experiments (random data) but… for real data… different story trivial solutions (rational solutions of f) still valid, but p correlated to d?
52p42d)128(3829 pdpd
Rational point on cone or conicRational point on cone or conic
Theory (Hasse) Q has solutions over the integers iff it
has solutions over the reals and mod every m (Legendre) Q has solutions modulo m for
every m not dividing 2abc (Holzer)
0: 222 czbyaxQ
||||,||||,|||| abzacybcx
Practice Efficient implementation (Cremona & Rusin,
2003) Modular solvers (msolve in Maple)
Re: Example from real dataRe: Example from real data
has canonical equationSQ 0222 zyx
[0,1,1,0] local [-15,4,1,1] global
,151215 22 vuvux ,124 22 vuy
,22 vuw ,z
432234 1052494280407 vuvvuvuu
ConclusionConclusion
Going for near-optimality is important simpler parameterizations smaller height of coefficients
Going for “full optimality” might be important for real data especially when the pencil has rational cones
More experiments needed
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