On Semiring Complexity of Schur Polynomials

Dima Grigoriev (Lille, CNRS)

jointly with S. Fomin, D. Nogneng, E. Schost

15/06/2017, Nice

Dima Grigoriev (CNRS) Complexity of Symmetric Functions 15.06.17 1 / 8

Elementary symmetric functionsek ,n =

∑1≤i1

Elementary symmetric functionsek ,n =

∑1≤i1

Elementary symmetric functionsek ,n =

∑1≤i1

Elementary symmetric functionsek ,n =

∑1≤i1

Elementary symmetric functionsek ,n =

∑1≤i1

Elementary symmetric functionsek ,n =

∑1≤i1

Elementary symmetric functionsek ,n =

∑1≤i1

Elementary symmetric functionsek ,n =

∑1≤i1

Monomial symmetric functions

For a partition λ = {λ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0} definemλ =

∑π∈Sn x

λ1π(1) · · · x

λnπ(n),

where the summation ranges over all the permutations π.Golomb ruler is a sequence of integers a1 > · · · > as such that thedifferences ai − aj , i < j are pairwise distinct.

Theorem

For a prime p the sequence λp−i+1 := 2pi + {i2 mod p}, 1 ≤ i ≤ p,where 0 ≤ i2 mod p < p, is a Golomb ruler (Erdös-Turán, 1941).

TheoremIf λ is a Golomb ruler then the complexity C+,×(mλ) ≥ Ω(cn), c > 1.(G.-Koshevoy).

Dima Grigoriev (CNRS) Complexity of Symmetric Functions 15.06.17 3 / 8

Monomial symmetric functions

For a partition λ = {λ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0} definemλ =

∑π∈Sn x

λ1π(1) · · · x

λnπ(n),

where the summation ranges over all the permutations π.Golomb ruler is a sequence of integers a1 > · · · > as such that thedifferences ai − aj , i < j are pairwise distinct.

Theorem

For a prime p the sequence λp−i+1 := 2pi + {i2 mod p}, 1 ≤ i ≤ p,where 0 ≤ i2 mod p < p, is a Golomb ruler (Erdös-Turán, 1941).

TheoremIf λ is a Golomb ruler then the complexity C+,×(mλ) ≥ Ω(cn), c > 1.(G.-Koshevoy).

Dima Grigoriev (CNRS) Complexity of Symmetric Functions 15.06.17 3 / 8

Monomial symmetric functions

For a partition λ = {λ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0} definemλ =

∑π∈Sn x

λ1π(1) · · · x

λnπ(n),

where the summation ranges over all the permutations π.Golomb ruler is a sequence of integers a1 > · · · > as such that thedifferences ai − aj , i < j are pairwise distinct.

Theorem

For a prime p the sequence λp−i+1 := 2pi + {i2 mod p}, 1 ≤ i ≤ p,where 0 ≤ i2 mod p < p, is a Golomb ruler (Erdös-Turán, 1941).

TheoremIf λ is a Golomb ruler then the complexity C+,×(mλ) ≥ Ω(cn), c > 1.(G.-Koshevoy).

Dima Grigoriev (CNRS) Complexity of Symmetric Functions 15.06.17 3 / 8

Monomial symmetric functions

For a partition λ = {λ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0} definemλ =

∑π∈Sn x

λ1π(1) · · · x

λnπ(n),

where the summation ranges over all the permutations π.Golomb ruler is a sequence of integers a1 > · · · > as such that thedifferences ai − aj , i < j are pairwise distinct.

Theorem

For a prime p the sequence λp−i+1 := 2pi + {i2 mod p}, 1 ≤ i ≤ p,where 0 ≤ i2 mod p < p, is a Golomb ruler (Erdös-Turán, 1941).

TheoremIf λ is a Golomb ruler then the complexity C+,×(mλ) ≥ Ω(cn), c > 1.(G.-Koshevoy).

Dima Grigoriev (CNRS) Complexity of Symmetric Functions 15.06.17 3 / 8

Monomial symmetric functions

For a partition λ = {λ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0} definemλ =

∑π∈Sn x

λ1π(1) · · · x

λnπ(n),

where the summation ranges over all the permutations π.Golomb ruler is a sequence of integers a1 > · · · > as such that thedifferences ai − aj , i < j are pairwise distinct.

Theorem

For a prime p the sequence λp−i+1 := 2pi + {i2 mod p}, 1 ≤ i ≤ p,where 0 ≤ i2 mod p < p, is a Golomb ruler (Erdös-Turán, 1941).

TheoremIf λ is a Golomb ruler then the complexity C+,×(mλ) ≥ Ω(cn), c > 1.(G.-Koshevoy).

Dima Grigoriev (CNRS) Complexity of Symmetric Functions 15.06.17 3 / 8

Lower bound on the complexity C+,×For a polynomial P denote by mon(P) the set of its monomials.

LemmaLet P be a homogeneous polynomial in n variables. If for anyhomogeneous polynomials Q,R such that mon(P) ⊃ mon(Q · R) and13 · deg(P) ≤ deg(Q), deg(R) ≤

23 · deg(P) we have

|mon(P)| > cn1 · |mon(Q · R)|, c1 > 1, then C+,×(P) ≥ Ω(cn2), c2 > 1.(Schnorr, 1976, Valiant, 1980)

LemmaLet λ be a Golomb ruler and let mon(Q · R) ⊂ mon(mλ) withhomogeneous Q,R. Then there is a subset S ⊂ {1, . . . ,n} such thatQ = Q′(xi , i ∈ S) ·M(xj , j 6∈ S), R = N(xi , i ∈ S) · R′(xj , j 6∈ S) whereM,N are monomials in variables xj , j 6∈ S and in xi , i ∈ S, respectively.(G.-Koshevoy).

Thus, one can separate the variables in such a way that Q,R dependon disjoint subsets of variables up to monomial factors.

Dima Grigoriev (CNRS) Complexity of Symmetric Functions 15.06.17 4 / 8

Lower bound on the complexity C+,×For a polynomial P denote by mon(P) the set of its monomials.

LemmaLet P be a homogeneous polynomial in n variables. If for anyhomogeneous polynomials Q,R such that mon(P) ⊃ mon(Q · R) and13 · deg(P) ≤ deg(Q), deg(R) ≤

23 · deg(P) we have

|mon(P)| > cn1 · |mon(Q · R)|, c1 > 1, then C+,×(P) ≥ Ω(cn2), c2 > 1.(Schnorr, 1976, Valiant, 1980)

LemmaLet λ be a Golomb ruler and let mon(Q · R) ⊂ mon(mλ) withhomogeneous Q,R. Then there is a subset S ⊂ {1, . . . ,n} such thatQ = Q′(xi , i ∈ S) ·M(xj , j 6∈ S), R = N(xi , i ∈ S) · R′(xj , j 6∈ S) whereM,N are monomials in variables xj , j 6∈ S and in xi , i ∈ S, respectively.(G.-Koshevoy).

Thus, one can separate the variables in such a way that Q,R dependon disjoint subsets of variables up to monomial factors.

Dima Grigoriev (CNRS) Complexity of Symmetric Functions 15.06.17 4 / 8

Lower bound on the complexity C+,×For a polynomial P denote by mon(P) the set of its monomials.

LemmaLet P be a homogeneous polynomial in n variables. If for anyhomogeneous polynomials Q,R such that mon(P) ⊃ mon(Q · R) and13 · deg(P) ≤ deg(Q), deg(R) ≤

23 · deg(P) we have

|mon(P)| > cn1 · |mon(Q · R)|, c1 > 1, then C+,×(P) ≥ Ω(cn2), c2 > 1.(Schnorr, 1976, Valiant, 1980)

LemmaLet λ be a Golomb ruler and let mon(Q · R) ⊂ mon(mλ) withhomogeneous Q,R. Then there is a subset S ⊂ {1, . . . ,n} such thatQ = Q′(xi , i ∈ S) ·M(xj , j 6∈ S), R = N(xi , i ∈ S) · R′(xj , j 6∈ S) whereM,N are monomials in variables xj , j 6∈ S and in xi , i ∈ S, respectively.(G.-Koshevoy).

Thus, one can separate the variables in such a way that Q,R dependon disjoint subsets of variables up to monomial factors.

Dima Grigoriev (CNRS) Complexity of Symmetric Functions 15.06.17 4 / 8

Lower bound on the complexity C+,×For a polynomial P denote by mon(P) the set of its monomials.

LemmaLet P be a homogeneous polynomial in n variables. If for anyhomogeneous polynomials Q,R such that mon(P) ⊃ mon(Q · R) and13 · deg(P) ≤ deg(Q), deg(R) ≤

23 · deg(P) we have

|mon(P)| > cn1 · |mon(Q · R)|, c1 > 1, then C+,×(P) ≥ Ω(cn2), c2 > 1.(Schnorr, 1976, Valiant, 1980)

LemmaLet λ be a Golomb ruler and let mon(Q · R) ⊂ mon(mλ) withhomogeneous Q,R. Then there is a subset S ⊂ {1, . . . ,n} such thatQ = Q′(xi , i ∈ S) ·M(xj , j 6∈ S), R = N(xi , i ∈ S) · R′(xj , j 6∈ S) whereM,N are monomials in variables xj , j 6∈ S and in xi , i ∈ S, respectively.(G.-Koshevoy).

Thus, one can separate the variables in such a way that Q,R dependon disjoint subsets of variables up to monomial factors.

Dima Grigoriev (CNRS) Complexity of Symmetric Functions 15.06.17 4 / 8

Lower bound on the complexity C+,×For a polynomial P denote by mon(P) the set of its monomials.

LemmaLet P be a homogeneous polynomial in n variables. If for anyhomogeneous polynomials Q,R such that mon(P) ⊃ mon(Q · R) and13 · deg(P) ≤ deg(Q), deg(R) ≤

23 · deg(P) we have

|mon(P)| > cn1 · |mon(Q · R)|, c1 > 1, then C+,×(P) ≥ Ω(cn2), c2 > 1.(Schnorr, 1976, Valiant, 1980)

LemmaLet λ be a Golomb ruler and let mon(Q · R) ⊂ mon(mλ) withhomogeneous Q,R. Then there is a subset S ⊂ {1, . . . ,n} such thatQ = Q′(xi , i ∈ S) ·M(xj , j 6∈ S), R = N(xi , i ∈ S) · R′(xj , j 6∈ S) whereM,N are monomials in variables xj , j 6∈ S and in xi , i ∈ S, respectively.(G.-Koshevoy).

Thus, one can separate the variables in such a way that Q,R dependon disjoint subsets of variables up to monomial factors.

Dima Grigoriev (CNRS) Complexity of Symmetric Functions 15.06.17 4 / 8

Lower bound on the complexity C+,×For a polynomial P denote by mon(P) the set of its monomials.

LemmaLet P be a homogeneous polynomial in n variables. If for anyhomogeneous polynomials Q,R such that mon(P) ⊃ mon(Q · R) and13 · deg(P) ≤ deg(Q), deg(R) ≤

23 · deg(P) we have

|mon(P)| > cn1 · |mon(Q · R)|, c1 > 1, then C+,×(P) ≥ Ω(cn2), c2 > 1.(Schnorr, 1976, Valiant, 1980)

LemmaLet λ be a Golomb ruler and let mon(Q · R) ⊂ mon(mλ) withhomogeneous Q,R. Then there is a subset S ⊂ {1, . . . ,n} such thatQ = Q′(xi , i ∈ S) ·M(xj , j 6∈ S), R = N(xi , i ∈ S) · R′(xj , j 6∈ S) whereM,N are monomials in variables xj , j 6∈ S and in xi , i ∈ S, respectively.(G.-Koshevoy).

Thus, one can separate the variables in such a way that Q,R dependon disjoint subsets of variables up to monomial factors.

Dima Grigoriev (CNRS) Complexity of Symmetric Functions 15.06.17 4 / 8

Lower bound on the complexity C+,×For a polynomial P denote by mon(P) the set of its monomials.

LemmaLet P be a homogeneous polynomial in n variables. If for anyhomogeneous polynomials Q,R such that mon(P) ⊃ mon(Q · R) and13 · deg(P) ≤ deg(Q), deg(R) ≤

23 · deg(P) we have

|mon(P)| > cn1 · |mon(Q · R)|, c1 > 1, then C+,×(P) ≥ Ω(cn2), c2 > 1.(Schnorr, 1976, Valiant, 1980)

LemmaLet λ be a Golomb ruler and let mon(Q · R) ⊂ mon(mλ) withhomogeneous Q,R. Then there is a subset S ⊂ {1, . . . ,n} such thatQ = Q′(xi , i ∈ S) ·M(xj , j 6∈ S), R = N(xi , i ∈ S) · R′(xj , j 6∈ S) whereM,N are monomials in variables xj , j 6∈ S and in xi , i ∈ S, respectively.(G.-Koshevoy).

Thus, one can separate the variables in such a way that Q,R dependon disjoint subsets of variables up to monomial factors.

Dima Grigoriev (CNRS) Complexity of Symmetric Functions 15.06.17 4 / 8

Lower bound on the complexity C+,×For a polynomial P denote by mon(P) the set of its monomials.

LemmaLet P be a homogeneous polynomial in n variables. If for anyhomogeneous polynomials Q,R such that mon(P) ⊃ mon(Q · R) and13 · deg(P) ≤ deg(Q), deg(R) ≤

23 · deg(P) we have

|mon(P)| > cn1 · |mon(Q · R)|, c1 > 1, then C+,×(P) ≥ Ω(cn2), c2 > 1.(Schnorr, 1976, Valiant, 1980)

LemmaLet λ be a Golomb ruler and let mon(Q · R) ⊂ mon(mλ) withhomogeneous Q,R. Then there is a subset S ⊂ {1, . . . ,n} such thatQ = Q′(xi , i ∈ S) ·M(xj , j 6∈ S), R = N(xi , i ∈ S) · R′(xj , j 6∈ S) whereM,N are monomials in variables xj , j 6∈ S and in xi , i ∈ S, respectively.(G.-Koshevoy).

Thus, one can separate the variables in such a way that Q,R dependon disjoint subsets of variables up to monomial factors.

Dima Grigoriev (CNRS) Complexity of Symmetric Functions 15.06.17 4 / 8

Complete symmetric polynomials

hk (x1, . . . , xn) :=∑

1≤i1≤···≤ik≤n xi1 · · · xikis homogeneous of degree k . Denote h̃k := hk (x21 , . . . , x

2n ). Then∑

0≤m

Complete symmetric polynomials

hk (x1, . . . , xn) :=∑

1≤i1≤···≤ik≤n xi1 · · · xikis homogeneous of degree k . Denote h̃k := hk (x21 , . . . , x

2n ). Then∑

0≤m

Complete symmetric polynomials

hk (x1, . . . , xn) :=∑

1≤i1≤···≤ik≤n xi1 · · · xikis homogeneous of degree k . Denote h̃k := hk (x21 , . . . , x

2n ). Then∑

0≤m

Complete symmetric polynomials

hk (x1, . . . , xn) :=∑

1≤i1≤···≤ik≤n xi1 · · · xikis homogeneous of degree k . Denote h̃k := hk (x21 , . . . , x

2n ). Then∑

0≤m

Complete symmetric polynomials

hk (x1, . . . , xn) :=∑

1≤i1≤···≤ik≤n xi1 · · · xikis homogeneous of degree k . Denote h̃k := hk (x21 , . . . , x

2n ). Then∑

0≤m

Complete symmetric polynomials

hk (x1, . . . , xn) :=∑

1≤i1≤···≤ik≤n xi1 · · · xikis homogeneous of degree k . Denote h̃k := hk (x21 , . . . , x

2n ). Then∑

0≤m

Complete symmetric polynomials

hk (x1, . . . , xn) :=∑

1≤i1≤···≤ik≤n xi1 · · · xikis homogeneous of degree k . Denote h̃k := hk (x21 , . . . , x

2n ). Then∑

0≤m

Complete symmetric polynomials

hk (x1, . . . , xn) :=∑

1≤i1≤···≤ik≤n xi1 · · · xikis homogeneous of degree k . Denote h̃k := hk (x21 , . . . , x

2n ). Then∑

0≤m

Complete symmetric polynomials

hk (x1, . . . , xn) :=∑

1≤i1≤···≤ik≤n xi1 · · · xikis homogeneous of degree k . Denote h̃k := hk (x21 , . . . , x

2n ). Then∑

0≤m

Complete symmetric polynomials

hk (x1, . . . , xn) :=∑

1≤i1≤···≤ik≤n xi1 · · · xikis homogeneous of degree k . Denote h̃k := hk (x21 , . . . , x

2n ). Then∑

0≤m

Complete symmetric polynomials

hk (x1, . . . , xn) :=∑

1≤i1≤···≤ik≤n xi1 · · · xikis homogeneous of degree k . Denote h̃k := hk (x21 , . . . , x

2n ). Then∑

0≤m

Schur polynomialsConsider a matrix with n rows and the infinite number of columnshaving (i , j)-entry equal x j−1i . For I = {i1 > · · · > in ≥ 1} denote by ∆Ithe n × n minor with the columns from I. In particular,∆n,n−1,...,1 =

∏i>j(xi − xj) is the Vandermonde minor.

For a partition λ = {λ1 ≥ · · · ≥ λn ≥ 0} the Schur polynomialsλ := ∆λ1+n,λ2+n−1,...,λn+1/∆n,n−1,...,1i. e. a symmetric polynomial with non-negative integer (Kostka)coefficients of the degree |λ| = λ1 + · · ·+ λn. Computing of Kostkacoefficients is #P-hard. Note that for λ = {λ1,0, . . . ,0} we havesλ = hλ1 . From the definition we get C+,−,×,/(sλ) ≤ O(n3 · log |λ|).

Theorem(V.Strassen, 1973). For any polynomial f it holdsC+,−,×(f ) ≤ O(C+,−,×,/(f ) · deg(f ) · log deg(f ))

Corollary

C+,−,×(sλ) ≤ O(n3 · |λ| · log2 |λ|).Dima Grigoriev (CNRS) Complexity of Symmetric Functions 15.06.17 6 / 8

Schur polynomialsConsider a matrix with n rows and the infinite number of columnshaving (i , j)-entry equal x j−1i . For I = {i1 > · · · > in ≥ 1} denote by ∆Ithe n × n minor with the columns from I. In particular,∆n,n−1,...,1 =

∏i>j(xi − xj) is the Vandermonde minor.

For a partition λ = {λ1 ≥ · · · ≥ λn ≥ 0} the Schur polynomialsλ := ∆λ1+n,λ2+n−1,...,λn+1/∆n,n−1,...,1i. e. a symmetric polynomial with non-negative integer (Kostka)coefficients of the degree |λ| = λ1 + · · ·+ λn. Computing of Kostkacoefficients is #P-hard. Note that for λ = {λ1,0, . . . ,0} we havesλ = hλ1 . From the definition we get C+,−,×,/(sλ) ≤ O(n3 · log |λ|).

Theorem(V.Strassen, 1973). For any polynomial f it holdsC+,−,×(f ) ≤ O(C+,−,×,/(f ) · deg(f ) · log deg(f ))

Corollary

C+,−,×(sλ) ≤ O(n3 · |λ| · log2 |λ|).Dima Grigoriev (CNRS) Complexity of Symmetric Functions 15.06.17 6 / 8

Schur polynomialsConsider a matrix with n rows and the infinite number of columnshaving (i , j)-entry equal x j−1i . For I = {i1 > · · · > in ≥ 1} denote by ∆Ithe n × n minor with the columns from I. In particular,∆n,n−1,...,1 =

∏i>j(xi − xj) is the Vandermonde minor.

For a partition λ = {λ1 ≥ · · · ≥ λn ≥ 0} the Schur polynomialsλ := ∆λ1+n,λ2+n−1,...,λn+1/∆n,n−1,...,1i. e. a symmetric polynomial with non-negative integer (Kostka)coefficients of the degree |λ| = λ1 + · · ·+ λn. Computing of Kostkacoefficients is #P-hard. Note that for λ = {λ1,0, . . . ,0} we havesλ = hλ1 . From the definition we get C+,−,×,/(sλ) ≤ O(n3 · log |λ|).

Theorem(V.Strassen, 1973). For any polynomial f it holdsC+,−,×(f ) ≤ O(C+,−,×,/(f ) · deg(f ) · log deg(f ))

Corollary

C+,−,×(sλ) ≤ O(n3 · |λ| · log2 |λ|).Dima Grigoriev (CNRS) Complexity of Symmetric Functions 15.06.17 6 / 8

Schur polynomialsConsider a matrix with n rows and the infinite number of columnshaving (i , j)-entry equal x j−1i . For I = {i1 > · · · > in ≥ 1} denote by ∆Ithe n × n minor with the columns from I. In particular,∆n,n−1,...,1 =

∏i>j(xi − xj) is the Vandermonde minor.

For a partition λ = {λ1 ≥ · · · ≥ λn ≥ 0} the Schur polynomialsλ := ∆λ1+n,λ2+n−1,...,λn+1/∆n,n−1,...,1i. e. a symmetric polynomial with non-negative integer (Kostka)coefficients of the degree |λ| = λ1 + · · ·+ λn. Computing of Kostkacoefficients is #P-hard. Note that for λ = {λ1,0, . . . ,0} we havesλ = hλ1 . From the definition we get C+,−,×,/(sλ) ≤ O(n3 · log |λ|).

Theorem(V.Strassen, 1973). For any polynomial f it holdsC+,−,×(f ) ≤ O(C+,−,×,/(f ) · deg(f ) · log deg(f ))

Corollary

C+,−,×(sλ) ≤ O(n3 · |λ| · log2 |λ|).Dima Grigoriev (CNRS) Complexity of Symmetric Functions 15.06.17 6 / 8

Schur polynomialsConsider a matrix with n rows and the infinite number of columnshaving (i , j)-entry equal x j−1i . For I = {i1 > · · · > in ≥ 1} denote by ∆Ithe n × n minor with the columns from I. In particular,∆n,n−1,...,1 =

∏i>j(xi − xj) is the Vandermonde minor.

For a partition λ = {λ1 ≥ · · · ≥ λn ≥ 0} the Schur polynomialsλ := ∆λ1+n,λ2+n−1,...,λn+1/∆n,n−1,...,1i. e. a symmetric polynomial with non-negative integer (Kostka)coefficients of the degree |λ| = λ1 + · · ·+ λn. Computing of Kostkacoefficients is #P-hard. Note that for λ = {λ1,0, . . . ,0} we havesλ = hλ1 . From the definition we get C+,−,×,/(sλ) ≤ O(n3 · log |λ|).

Theorem(V.Strassen, 1973). For any polynomial f it holdsC+,−,×(f ) ≤ O(C+,−,×,/(f ) · deg(f ) · log deg(f ))

Corollary

C+,−,×(sλ) ≤ O(n3 · |λ| · log2 |λ|).Dima Grigoriev (CNRS) Complexity of Symmetric Functions 15.06.17 6 / 8

Schur polynomialsConsider a matrix with n rows and the infinite number of columnshaving (i , j)-entry equal x j−1i . For I = {i1 > · · · > in ≥ 1} denote by ∆Ithe n × n minor with the columns from I. In particular,∆n,n−1,...,1 =

∏i>j(xi − xj) is the Vandermonde minor.

For a partition λ = {λ1 ≥ · · · ≥ λn ≥ 0} the Schur polynomialsλ := ∆λ1+n,λ2+n−1,...,λn+1/∆n,n−1,...,1i. e. a symmetric polynomial with non-negative integer (Kostka)coefficients of the degree |λ| = λ1 + · · ·+ λn. Computing of Kostkacoefficients is #P-hard. Note that for λ = {λ1,0, . . . ,0} we havesλ = hλ1 . From the definition we get C+,−,×,/(sλ) ≤ O(n3 · log |λ|).

Theorem(V.Strassen, 1973). For any polynomial f it holdsC+,−,×(f ) ≤ O(C+,−,×,/(f ) · deg(f ) · log deg(f ))

Corollary

C+,−,×(sλ) ≤ O(n3 · |λ| · log2 |λ|).Dima Grigoriev (CNRS) Complexity of Symmetric Functions 15.06.17 6 / 8

Schur polynomialsConsider a matrix with n rows and the infinite number of columnshaving (i , j)-entry equal x j−1i . For I = {i1 > · · · > in ≥ 1} denote by ∆Ithe n × n minor with the columns from I. In particular,∆n,n−1,...,1 =

∏i>j(xi − xj) is the Vandermonde minor.

For a partition λ = {λ1 ≥ · · · ≥ λn ≥ 0} the Schur polynomialsλ := ∆λ1+n,λ2+n−1,...,λn+1/∆n,n−1,...,1i. e. a symmetric polynomial with non-negative integer (Kostka)coefficients of the degree |λ| = λ1 + · · ·+ λn. Computing of Kostkacoefficients is #P-hard. Note that for λ = {λ1,0, . . . ,0} we havesλ = hλ1 . From the definition we get C+,−,×,/(sλ) ≤ O(n3 · log |λ|).

Theorem(V.Strassen, 1973). For any polynomial f it holdsC+,−,×(f ) ≤ O(C+,−,×,/(f ) · deg(f ) · log deg(f ))

Corollary

C+,−,×(sλ) ≤ O(n3 · |λ| · log2 |λ|).Dima Grigoriev (CNRS) Complexity of Symmetric Functions 15.06.17 6 / 8

Schur polynomialsConsider a matrix with n rows and the infinite number of columnshaving (i , j)-entry equal x j−1i . For I = {i1 > · · · > in ≥ 1} denote by ∆Ithe n × n minor with the columns from I. In particular,∆n,n−1,...,1 =

∏i>j(xi − xj) is the Vandermonde minor.

For a partition λ = {λ1 ≥ · · · ≥ λn ≥ 0} the Schur polynomialsλ := ∆λ1+n,λ2+n−1,...,λn+1/∆n,n−1,...,1i. e. a symmetric polynomial with non-negative integer (Kostka)coefficients of the degree |λ| = λ1 + · · ·+ λn. Computing of Kostkacoefficients is #P-hard. Note that for λ = {λ1,0, . . . ,0} we havesλ = hλ1 . From the definition we get C+,−,×,/(sλ) ≤ O(n3 · log |λ|).

Theorem(V.Strassen, 1973). For any polynomial f it holdsC+,−,×(f ) ≤ O(C+,−,×,/(f ) · deg(f ) · log deg(f ))

Corollary

C+,−,×(sλ) ≤ O(n3 · |λ| · log2 |λ|).Dima Grigoriev (CNRS) Complexity of Symmetric Functions 15.06.17 6 / 8

Schur polynomialsConsider a matrix with n rows and the infinite number of columnshaving (i , j)-entry equal x j−1i . For I = {i1 > · · · > in ≥ 1} denote by ∆Ithe n × n minor with the columns from I. In particular,∆n,n−1,...,1 =

∏i>j(xi − xj) is the Vandermonde minor.

For a partition λ = {λ1 ≥ · · · ≥ λn ≥ 0} the Schur polynomialsλ := ∆λ1+n,λ2+n−1,...,λn+1/∆n,n−1,...,1i. e. a symmetric polynomial with non-negative integer (Kostka)coefficients of the degree |λ| = λ1 + · · ·+ λn. Computing of Kostkacoefficients is #P-hard. Note that for λ = {λ1,0, . . . ,0} we havesλ = hλ1 . From the definition we get C+,−,×,/(sλ) ≤ O(n3 · log |λ|).

Theorem(V.Strassen, 1973). For any polynomial f it holdsC+,−,×(f ) ≤ O(C+,−,×,/(f ) · deg(f ) · log deg(f ))

Corollary

C+,−,×(sλ) ≤ O(n3 · |λ| · log2 |λ|).Dima Grigoriev (CNRS) Complexity of Symmetric Functions 15.06.17 6 / 8

Schur polynomialsConsider a matrix with n rows and the infinite number of columnshaving (i , j)-entry equal x j−1i . For I = {i1 > · · · > in ≥ 1} denote by ∆Ithe n × n minor with the columns from I. In particular,∆n,n−1,...,1 =

∏i>j(xi − xj) is the Vandermonde minor.

For a partition λ = {λ1 ≥ · · · ≥ λn ≥ 0} the Schur polynomialsλ := ∆λ1+n,λ2+n−1,...,λn+1/∆n,n−1,...,1i. e. a symmetric polynomial with non-negative integer (Kostka)coefficients of the degree |λ| = λ1 + · · ·+ λn. Computing of Kostkacoefficients is #P-hard. Note that for λ = {λ1,0, . . . ,0} we havesλ = hλ1 . From the definition we get C+,−,×,/(sλ) ≤ O(n3 · log |λ|).

Theorem(V.Strassen, 1973). For any polynomial f it holdsC+,−,×(f ) ≤ O(C+,−,×,/(f ) · deg(f ) · log deg(f ))

Corollary

C+,−,×(sλ) ≤ O(n3 · |λ| · log2 |λ|).Dima Grigoriev (CNRS) Complexity of Symmetric Functions 15.06.17 6 / 8

Complexity of Schur polynomials

Theorem

(S.Fomin-G.-Koshevoy). C+,×,/(sλ) ≤ O((n + |λ|)3).

The proof is based on the cluster transformations.

Theorem

(Demmel-Koev). C+,×(sλ) ≤ O(exp(|λ|) · n2).

Theorem

(S.Fomin-G.-Nogneng-Schost). C+,×(sλ) ≤ O(exp(n2) · log |λ|).

Question. C+,×(sλ) ≤ (n + |λ|)O(1)?

Dima Grigoriev (CNRS) Complexity of Symmetric Functions 15.06.17 7 / 8

Complexity of Schur polynomials

Theorem

(S.Fomin-G.-Koshevoy). C+,×,/(sλ) ≤ O((n + |λ|)3).

The proof is based on the cluster transformations.

Theorem

(Demmel-Koev). C+,×(sλ) ≤ O(exp(|λ|) · n2).

Theorem

(S.Fomin-G.-Nogneng-Schost). C+,×(sλ) ≤ O(exp(n2) · log |λ|).

Question. C+,×(sλ) ≤ (n + |λ|)O(1)?

Dima Grigoriev (CNRS) Complexity of Symmetric Functions 15.06.17 7 / 8

Complexity of Schur polynomials

Theorem

(S.Fomin-G.-Koshevoy). C+,×,/(sλ) ≤ O((n + |λ|)3).

The proof is based on the cluster transformations.

Theorem

(Demmel-Koev). C+,×(sλ) ≤ O(exp(|λ|) · n2).

Theorem

(S.Fomin-G.-Nogneng-Schost). C+,×(sλ) ≤ O(exp(n2) · log |λ|).

Question. C+,×(sλ) ≤ (n + |λ|)O(1)?

Dima Grigoriev (CNRS) Complexity of Symmetric Functions 15.06.17 7 / 8

Complexity of Schur polynomials

Theorem

(S.Fomin-G.-Koshevoy). C+,×,/(sλ) ≤ O((n + |λ|)3).

The proof is based on the cluster transformations.

Theorem

(Demmel-Koev). C+,×(sλ) ≤ O(exp(|λ|) · n2).

Theorem

(S.Fomin-G.-Nogneng-Schost). C+,×(sλ) ≤ O(exp(n2) · log |λ|).

Question. C+,×(sλ) ≤ (n + |λ|)O(1)?

Dima Grigoriev (CNRS) Complexity of Symmetric Functions 15.06.17 7 / 8

Complexity of Schur polynomials

Theorem

(S.Fomin-G.-Koshevoy). C+,×,/(sλ) ≤ O((n + |λ|)3).

The proof is based on the cluster transformations.

Theorem

(Demmel-Koev). C+,×(sλ) ≤ O(exp(|λ|) · n2).

Theorem

(S.Fomin-G.-Nogneng-Schost). C+,×(sλ) ≤ O(exp(n2) · log |λ|).

Question. C+,×(sλ) ≤ (n + |λ|)O(1)?

Dima Grigoriev (CNRS) Complexity of Symmetric Functions 15.06.17 7 / 8

Complexity of Schur polynomials

Theorem

(S.Fomin-G.-Koshevoy). C+,×,/(sλ) ≤ O((n + |λ|)3).

The proof is based on the cluster transformations.

Theorem

(Demmel-Koev). C+,×(sλ) ≤ O(exp(|λ|) · n2).

Theorem

(S.Fomin-G.-Nogneng-Schost). C+,×(sλ) ≤ O(exp(n2) · log |λ|).

Question. C+,×(sλ) ≤ (n + |λ|)O(1)?

Dima Grigoriev (CNRS) Complexity of Symmetric Functions 15.06.17 7 / 8

Semi-standard Young tableaux

The proof of the bound on C+,×(sλ) relies on the obtained bound onC+,×(hk ) and on the formula expressing sλ via the semi-standardYoung tableaux T with the shape λ:T = (ti,j), 1 ≤ ti,j ≤ n, 1 ≤ i ≤ n, 1 ≤ j ≤ λi ,ti+1,j > ti,j (strictly increasing down) andti,j+1 ≥ ti,j (non-decreasing to the right).Monomial xT :=

∏i,j∈T xti,j . Then sλ =

∑T x

T where the summationranges over all the semi-standard tableaux with the shape λ.

Dima Grigoriev (CNRS) Complexity of Symmetric Functions 15.06.17 8 / 8

Semi-standard Young tableaux

The proof of the bound on C+,×(sλ) relies on the obtained bound onC+,×(hk ) and on the formula expressing sλ via the semi-standardYoung tableaux T with the shape λ:T = (ti,j), 1 ≤ ti,j ≤ n, 1 ≤ i ≤ n, 1 ≤ j ≤ λi ,ti+1,j > ti,j (strictly increasing down) andti,j+1 ≥ ti,j (non-decreasing to the right).Monomial xT :=

∏i,j∈T xti,j . Then sλ =

∑T x

T where the summationranges over all the semi-standard tableaux with the shape λ.

Dima Grigoriev (CNRS) Complexity of Symmetric Functions 15.06.17 8 / 8

Semi-standard Young tableaux

The proof of the bound on C+,×(sλ) relies on the obtained bound onC+,×(hk ) and on the formula expressing sλ via the semi-standardYoung tableaux T with the shape λ:T = (ti,j), 1 ≤ ti,j ≤ n, 1 ≤ i ≤ n, 1 ≤ j ≤ λi ,ti+1,j > ti,j (strictly increasing down) andti,j+1 ≥ ti,j (non-decreasing to the right).Monomial xT :=

∏i,j∈T xti,j . Then sλ =

∑T x

T where the summationranges over all the semi-standard tableaux with the shape λ.

Dima Grigoriev (CNRS) Complexity of Symmetric Functions 15.06.17 8 / 8

Semi-standard Young tableaux

The proof of the bound on C+,×(sλ) relies on the obtained bound onC+,×(hk ) and on the formula expressing sλ via the semi-standardYoung tableaux T with the shape λ:T = (ti,j), 1 ≤ ti,j ≤ n, 1 ≤ i ≤ n, 1 ≤ j ≤ λi ,ti+1,j > ti,j (strictly increasing down) andti,j+1 ≥ ti,j (non-decreasing to the right).Monomial xT :=

∏i,j∈T xti,j . Then sλ =

∑T x

T where the summationranges over all the semi-standard tableaux with the shape λ.

Dima Grigoriev (CNRS) Complexity of Symmetric Functions 15.06.17 8 / 8

Semi-standard Young tableaux

The proof of the bound on C+,×(sλ) relies on the obtained bound onC+,×(hk ) and on the formula expressing sλ via the semi-standardYoung tableaux T with the shape λ:T = (ti,j), 1 ≤ ti,j ≤ n, 1 ≤ i ≤ n, 1 ≤ j ≤ λi ,ti+1,j > ti,j (strictly increasing down) andti,j+1 ≥ ti,j (non-decreasing to the right).Monomial xT :=

∏i,j∈T xti,j . Then sλ =

∑T x

T where the summationranges over all the semi-standard tableaux with the shape λ.

Dima Grigoriev (CNRS) Complexity of Symmetric Functions 15.06.17 8 / 8

Semi-standard Young tableaux

The proof of the bound on C+,×(sλ) relies on the obtained bound onC+,×(hk ) and on the formula expressing sλ via the semi-standardYoung tableaux T with the shape λ:T = (ti,j), 1 ≤ ti,j ≤ n, 1 ≤ i ≤ n, 1 ≤ j ≤ λi ,ti+1,j > ti,j (strictly increasing down) andti,j+1 ≥ ti,j (non-decreasing to the right).Monomial xT :=

∏i,j∈T xti,j . Then sλ =

∑T x

T where the summationranges over all the semi-standard tableaux with the shape λ.

Dima Grigoriev (CNRS) Complexity of Symmetric Functions 15.06.17 8 / 8

Semi-standard Young tableaux

The proof of the bound on C+,×(sλ) relies on the obtained bound onC+,×(hk ) and on the formula expressing sλ via the semi-standardYoung tableaux T with the shape λ:T = (ti,j), 1 ≤ ti,j ≤ n, 1 ≤ i ≤ n, 1 ≤ j ≤ λi ,ti+1,j > ti,j (strictly increasing down) andti,j+1 ≥ ti,j (non-decreasing to the right).Monomial xT :=

∏i,j∈T xti,j . Then sλ =

∑T x

T where the summationranges over all the semi-standard tableaux with the shape λ.

Dima Grigoriev (CNRS) Complexity of Symmetric Functions 15.06.17 8 / 8