(Uniform) determinantal representations

1 (Uniform) determinantal representations Jan Draisma Universität Bern October 216, Kolloquium Bern

Determinantal representations 2 R := C[x 1,..., x n ] and R d := {p R deg p d} Definition A determinantal representation of p R of size N is a matrix M R N N 1 with det(m) = p.

Determinantal representations 2 R := C[x 1,..., x n ] and R d := {p R deg p d} Definition A determinantal representation of p R of size N is a matrix M R N N 1 with det(m) = p. n = 1: companion matrices x 1 x 1 det...... x 1 a a 1 a n 2 a n 1 + a n x = a + a 1 x +... + a n x n

Determinantal representations 2 R := C[x 1,..., x n ] and R d := {p R deg p d} Definition A determinantal representation of p R of size N is a matrix M R N N 1 with det(m) = p. A bivariate example x 1 det y 1 a + bx + cy dx + ey f y = a + bx + cy + dx2 + exy + f y 2

Why? 3

Motivation I: permanent versus determinant 4 If p has a determinantal representation M of small size N, then p can be evaluated efficiently using Gaussian elimination.

Motivation I: permanent versus determinant 4 If p has a determinantal representation M of small size N, then p can be evaluated efficiently using Gaussian elimination. Definition perm m := π S m x 1π(1) x mπ(m) is the m m permanent. Example 1 1 perm 3 1 1 1 1 1 = 3 counts perfect matchings:

Geometric complexity theory 5 Conjecture dc(perm m ) grows faster with m than any polynomial. [Valiant, 7s]

Geometric complexity theory 5 Conjecture dc(perm m ) grows faster with m than any polynomial. [Valiant, 7s] Best known bounds [Mignon-Ressayre 4, Grenet 12] m 2 2 dc(perm m ) 2m 1 [Alper-Bogart-Velasco 15: = 7 for m = 3] Proof sketch of lower bound If ψ : C m m C N N affine-linear with det N (ψ(a)) = perm m (A), J := m + 1 1 1 1 1 1... ψ(j) 1 1 1 perm m = det N = q 1 (X) := quadratic part of perm m (J + X), form of rank m 2 q 2 (Y) := quadratic part of det N (ψ(j) + Y), form of rank 2N J

Grenet s 2 m 1 construction 6 = 123 x 11 x 12 x 13 1 2 3 x 22 x 23 12 x 21 x 23 x 21 x 22 13 x 33 x 32 x 31 23 = 123 1 2 3 12 13 23 = 123 1 2 3 12 13 23 x 11 x 12 x 13 1 x 22 x 23 1 x 21 x 23 1 x 21 x 22 x 33 1 x 32 1 x 31 1 123 = x i j labels an arrow from an (i 1)-set to an i-set by adding j.

Geometric complexity theory 7 Theorem [Landsberg-Ressayre, 15] Grenet s representation is optimal among representations that preserve left multiplication with permutation and diagonal matrices.

Geometric complexity theory 7 Theorem [Landsberg-Ressayre, 15] Grenet s representation is optimal among representations that preserve left multiplication with permutation and diagonal matrices. GCT Programme [Mulmuley-Sohoni, 1-] Compare orbit closures X 1, X 2 of l N m perm m and det N inside the space of degree-n polynomials in N 2 variables under G = GL N 2; try to show that X 1 X 2 by showing that multiplicities of certain G-representations are higher in C[X 1 ] than in C[X 2 ] unless N is super-polynomial in m.

Motivation II: Solving systems of equations 8 In numerics, solving a univariate equation p(x) = is often done by finding the eigenvalues of the companion matrix of p.

Motivation II: Solving systems of equations 8 In numerics, solving a univariate equation p(x) = is often done by finding the eigenvalues of the companion matrix of p. Proposal [Plestenjak-Hochstenbach, 16] To solve p(x, y) = q(x, y) = write p = det(a + xa 1 + ya 2 ) and q = det(b + xb 1 x + yb 2 ) and solve the two-parameter eigenvalue problem (A + xa 1 + ya 2 )u = and (B + xb 1 + yb 2 )v =. translates to a joint pair of generalised eigenvalue problems: ( 1 x )w = and ( 2 y )w = where w = u v and = A 1 B 2 A 2 B 1, 1 = A 2 B A B 2, 2 = A B 1 A 1 B

Determinantal complexity of general polynomials 9 Theorem [Boralevi-v Doornmalen-D-Hochstenbach-Plestenjak, 16] For n fixed, there exist C 1, C 2 such that a sufficiently general p R d has dc(p) C 1 d n/2 and any p R d has dc(p) C 2 d n/2. For the upper bound, the determinantal representation can be chosen to depend bi-affine-linearly on x 1,..., x n and on the coefficients of p; these are uniform determinantal representations. Proof of lower bound If sufficiently general p R d have dc(p) N, then the map det : R N N 1 R N contains R d in the closure of its image. Comparing ( ) n + d dimensions, find N 2 (n + 1) dim C R d =. n

Spaces connected to 1 1 Definition Given a nonzero subspace V R write V d := V R d. V is connected to 1 if V d+1 R 1 V d for all d.

Spaces connected to 1 1 Definition Given a nonzero subspace V R write V d := V R d. V is connected to 1 if V d+1 R 1 V d for all d. Example For n = 2, V spanned by these monomials is connected to 1: 1 x y x 2 xy y 2 x 3 x 2 y xy 2 y 3 x 4 x 3 y x 2 y 2 xy 3 y 4 Lemma V connected to 1, with basis 1 = f 1, f 2,..., f m of ascending degrees, write f i = i 1 j=1 l i j f j with l i j R 1. Then V = the span of the (m 1) (m 1)-subdeterminants of M(V) := l 21 1 l 31 l 32 1....... l m1 l m2 l m,m 1 1

First construction 11 Proposition Let V R be connected to 1, of dimension m, and such that R 1 V R d. Then there is a uniform determinantal representation of size m for the polynomials in R d.

First construction 11 Proposition Let V R be connected to 1, of dimension m, and such that R 1 V R d. Then there is a uniform determinantal representation of size m for the polynomials in R d. Example x 1 det y 1 a + bx + cy dx + ey f y = a + bx + cy + dx 2 + exy + f y 2 M(V) V : 1 x y x 2 xy y 2

Analysis of first construction 12 V connected to 1 and R 1 V R d imply dim V 1 n ( n + d n )

Analysis of first construction 12 V connected to 1 and R 1 V R d imply dim V 1 n ( n + d n ) Proposition For fixed n, uniform determinantal representation of size dn n n!.

Analysis of first construction 12 V connected to 1 and R 1 V R d imply dim V 1 n ( n + d n ) Proposition For fixed n, uniform determinantal representation of size dn n n!. Construction uses the lattice of type A n 1 with generating matrix 2 1. 1 2........ 1 1 2 (David Madore, YouTube)

Second construction: divide and conquer! 13 Proposition Suppose V 1, V 2 R connected to 1 such that R 1 V 1 V 2 R d. Then there is a uniform det representation of degree-d polynomials of size 1 + dim V 1 + dim V 2. M(V 1 ) Example x 1 x 1 det c c 1 c 2 y = i+ j 2 c i j x i y j c 1 c 11 1 y M(V c 2 1 2 ) T

A fractal 14 Can we find V 1, V 2, connected to 1, of dim growing like dim R d such that (R 1 )V 1 V 2 R d?

A fractal 14 Can we find V 1, V 2, connected to 1, of dim growing like dim R d such that (R 1 )V 1 V 2 R d? ( ) n/2 + d For n even, split variables V 1, V 2 of dimension. n/2

A fractal 14 Can we find V 1, V 2, connected to 1, of dim growing like dim R d such that (R 1 )V 1 V 2 R d? ( ) n/2 + d For n even, split variables V 1, V 2 of dimension. n/2 For odd n, find subsets A, A 1 (Z ) n, connected to, of dimension n 2 such that A + A 1 = Z n : - start with B := j= {, 1} 2 2 j ; - B 1 := 2B so that B + B 1 = Z ; - A i := B n i ; - connect to.

Outlook 15 Theorem [Boralevi-v Doornmalen-D-Hochstenbach-Plestenjak, 16] For n fixed, there exist C 1, C 2 such that a sufficiently general p R d has dc(p) C 1 d n/2 and any p R d has dc(p) C 2 d n/2. Many questions remain: what are the best constants C 1, C 2? what about the regime where d is fixed and n runs? symmetric determinantal representations? Thank you! Motivation III: hyperbolic polynomials If p = det(a + i x i A i ) with A i R N N symmetric and A positive definite, then the restriction of p to any line through has only real roots. For n = 2 the converse also holds (Helton-Vinnikov).