Proving Algebraic Identities Avi Wigderson IAS, Princeton

1. Proving Algebraic Identities Avi Wigderson IAS, Princeton Math and Computation New book on my website

2. Plan • Algebraic identities & word problems • Arithmetic computational complexity • Representing identities: Symbolic matrices • Non-commutative singularity • NC algebra, free skew fields, word problem • Invariant theory: nullcone, degree bounds

3. Algebraic identities 1 x11 x12 …x1n-1 1 x21 x22 …x2n-1 …………………….. 1 xn1 xn2 …xnn-1 F field (large), xi variables [Vandermonde,1771] How can we verify such an identity efficiently? Numerous across math, proven & conjectured Expanding the determinant has exponentially many terms! General problem: given p  F[x1,x2,…xn], is p=0? given r  F(x1,x2,…xn), is r=0? Word problem: are two representations ``equivalent’’ ? det = ∏i<j (xi-xj)

4. Arithmetic Complexity

5. Representing polynomials p= (x1+y1)(x2+y2)…(xn+yn) + × …… exp(n) + + × × + × x2 x2 y2 y1 y2 yn xn xn yn x1 y1 y1 x1 x2 xn …… poly(n)

6. p p + + ÷ F field × × n variables, deg p <nc × + Algebraic complexityrepresenting polynomials (& rational functions) + + + × c × X5 Xi Xi Xj Xj Xi Xi c’ c Formula L(p) – formula size Circuit S(p) – Circuit size  [VSBR]: S(p) ≤ L(p) ≤ S(p)logn VP = { p:S(p) ≤ poly(n) } Easy polynomials

7. VP = VNP ? [Valiant’79] Fd[x1,x2,…xn] * Permanent * Random Enumeration polynomials * Determinant * Matrix Mult * DFT *Sym VNP:“interesting”, explicit polynomials VP: easy polynomials Stat. Physics polynomials Can we efficiently compute everything we care about?

8. VP = VNP ? [Valiant’79] char(F)2. XMn(F) matrix of variables xij Detn(X) = Snsgn() i[n] xi(i) Pern(X) = Sni[n] xi(i) Homogeneous, multi-linear, degree n polynomials on n2 variables, with 0,±1 coefficients. [Valiant’79]Det: Easy(!), complete for VP Per: Hard(?), complete for VNP

9. [Gauss+]: DetVP:S(Detn) ≤n3 (no division!) [Valiant]: Det is complete for VP Why does Det appear all over Mathematics? Jacobian, Wronskian, Vandermodian, Pfaffian,… Every arith formula is a symbolic determinant! [Valiant]: Every small arithmetic formula is a small symbolic determinant. If L(f)=s, then there is a 2s×2s symbolic matrix Mf with f=det Mf Complexity of Det

10. [Valiant]: If L(f)=s, then there is a 2s×2s symbolic matrix Mf with f=det Mf Proof: Induction f=g+h f=g×h 1 0 1 0 0 1 Completeness of Det Stronger 1 1 Mg 1 Mf 1 0 1 0 0 1 Mg Mf 1 0 1 0 0 1 0 0 Mh 1 1 Mh 1 1 0 1 1 0 1 x MX 1 0 1 0 0 |Mf|=|Mg|+|Mh| |Mf|=|Mg|×|Mh|

11. VP≠VNP? L11 L12 L13 L21 L22 L23 Does VPVNP?  Does Per have small formula?  Does Per have small symbolic determinant? Pern(X) = Detk( )Lij(X) affine, ksmall? L31 L32 L33

12. Proving VP≠VNP Affine mapL: Mn(F)  Mk(F) is good if Pern = Detk L k(n): the smallest k for which there is a goodmap [Polya] k(2) =2 Per2 = Det2 k(3)>3 [Valiant] k(n) < exp(n) [Valiant] k(n)  poly(n) VPVNP [Mignon-Ressayre] k(n) > n2 Approaches: * Rank methods (flattenings) * Geometric Complexity Theory * De-randomization of PIT (Polynomial Identities) a b c d a b -c d

13. Proving polynomial identities

14. Polynomial Identity Testing (PIT)(Word Problem) Given: p  F[x1,x2,…xn] (as formula, circuit or symbolic determinant) Problem: Is p=0? [DL,S,Z,…] PIT has an efficientprobabilistic algorithm Pick at random a=(a1,a2,…an), ai {1,2,…,2deg(p)} iid If p=0, p(a)=0 with probability = 1 If p≠0, p(a)=0 with probability < ½ De-randomization: find efficientdeterministic algorithm? [Kabanets-Impagliazzo] efficientdeterministic algorithm (for proving algebraic identities)  “VPVNP” Proof: Pseudorandomness, Diagonalization

15. How to solve PIT? [KI’01] efficientdeterministic PIT algorithm  “VPVNP” Use assumptions/Change the problem/Weaken the problem [KI’01] VPVNP  efficientdeterministic PIT algorithm [DS,KS,…,…,…] Program:PIT for “simple” formulae A+B+C=0? A=∏iai,B=∏ibi,C=∏ici,ai,bi,ci F[x1…xn] linear (A+B=0?Easy: unique factorization) [DS] dim{ai,bi,ci} < O(log n) Error correcting codes [KS] dim{ai,bi,ci} < O(1) Combinatorial geometry “Observe”: Setting any ai=bj=0 sets some ck=0 Many co-linear triples  low dimension

16. [Sylvester-Gallai] - Finite collection of points in Rd Line through every pair of points hits a 3rd  They are all on one line! (dim=1)

17. X = {x1,x2,… xm} F field A(X) = A1x1+A2x2+…+Amxm Input: A1,A2,…,Am Mn(F) SING: Is A(X) singular? What we want to solve: xi commute in F(x1,x2,…,xm) [Edmonds’67] SING P? [Lovasz ’79] SINGRP L11 L12 L13 L21 L22 L23 PIT: symbolic matrices singularity L31 L32 L33 What we do solve: xi do not commute in F<(x1,x2,…,xm)> (free skew field) [Cohn’75] NC-SINGdecidable [CR’99] NC-SINGEXP [GGOW’15] NC-SING P (F=Q) [IQS’16] NC-SINGP (F large)

18. Non-commutative algebra

19. X = {x1,x2,…} non-commutative, F (commutative) field F<X> polynomials, e.g. p(X) = 1+ xy+yx F<(X)> rational expressions (= formulas), e.g. r(X) = x-1+y-1 r(X) = (x+zy-1w)-1 [Reutenauer’96] Unbounded nested inversion r(X) = (x+xy-1x)-1 r(X) = 0 ? Word Problem [Garg-Gurvits-Oliveira-W’15] In P char(F)=0 [Ivanyos-Qiao-Subrahmanyam’16] In P F large Non-commutative identitiesWord problem for free skew fields notpq-1 orp-1q Nested inversion = (x+y)-1- x-1 Hua’s identity

20. X = {x1,x2,…} non-commutative, F (commutative) field F<X> polynomials, e.g. p(X) = 1+ xy+yx F<(X)> rational expressions, e.g. r(X) = x-1 +y-1 r(X) = 0 ? Word Problem in P [Cohn’71] Rational expression  symbolic matrix inverse r A1,A2,…,Am Mn(F) m,n ≤ |r| such that r(x1,x2,…,xm)=0  A(X)=A1x1+A2x2+…+Amxm NC-SING [Amitsur’61]  Det(i AiDi) = 0 d, Di  Md(F)  Det(i AiXi) = 0 d, Xi  Md(F[X]) Infinitely many (commutative) algebraic identities! Bound d? NC-SING is a group invariant! Non-commutative algebraWord problem for free skew fields Generalizing [Valiant]

21. R Invariant Theorysymmetries, group actions,orbits, invariants - Energy - Momentum 1 1 5 2 5 2 S5 4 3 4 3 ??? Area

22. G acts linearly on V=Fk ,and so on F[z1,z2,…,zk] VG= { p F[z]: p(gz) = p(z) for all g  G} Ex1: G=Snacts on V=Fn by permuting coordinates VG=< elementary symmetric polynomials > Ex2: G=SLn(F)2acts on V=Mn(F) by Z RZC VG=< det(Z) > Left-Right action (generalizes Ex2): A=(A1,A2,…,Am)  Mn(F)m = V. G=SLn(F)2acts on V by simultaneous basis changesR,C A RAC =(RA1C,RA2C,…,RAmC) Invariant ring Invariant theoryLinear actions Generators: few, low degree, easily computable

23. A=(A1,A2,…,Am)  Mn(F)m =V G=SLn(F)2acts on V:RAC =(RA1C,RA2C,…,RAmC) (Zi)jkmn2(commuting) vars F[Z]G={p polynomial: p(RZC) = p(Z) for all R,C  SLn(F)} [DW,DZ,SV’00] F[Z]G = <det(i ZiDi) : d N, Di Md(F)> Nullcone: Given A, does p(A)=0 for all pF[Z]G ?  ANC-SING Degree bounds [Hilbert’90] d< ∞F[Z]G finitely generated [Popov’81] d< exp(exp(n)) [Derksen’01] d< exp(n)  [GGOW’15] (capacity analysis) [DM’15] d< poly(n)  [IQS’16] (combinatorial alg.) Proof essentially uses Non-Commututative algebra! Invariant theory Left-Right action In P

24. Summary • Algebraic identities & word problems • Symbolic determinant & inverse: completeness • NC-SING: solved (Commutative Invariant Th., Quantum Inf. Th.) • C-SING:open (de-randomization,VPVNP) • Is C-SING a group invariant? • New efficient algorithms in Invariant Theory (nullcone, orbit closure intersection, other actions)