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

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

Algebraic identities = ∏i<j (xi-xj) det F field (large), 1 x11 x12 … x1n-1 1 x21 x22 … x2n-1 …………………….. 1 xn1 xn2 … xnn-1 = ∏i<j (xi-xj) det 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’’ ?

Arithmetic Complexity

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

Algebraic complexity representing polynomials (& rational functions) + × Xi Xj c’ p c X5 × + Xi Xj c p F field ÷ n variables, deg p <nc 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

VP = VNP ? [Valiant’79] Fd[x1,x2,…xn] VNP:“interesting”, VP: easy * 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?

VP = VNP ? [Valiant’79] char(F)2. XMn(F) matrix of variables xij Detn(X) = Sn sgn() 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

Complexity of Det [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 Determinantal representations of polys

Completeness of Det [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 Stronger 1 1 0 1 0 0 1 Mg Mh Mf Determinantal representations of polys Mg 1 0 1 0 0 1 Mf 1 Mh 1 0 1 MX 1 0 1 x |Mf|=|Mg|+|Mh| |Mf|=|Mg|×|Mh|

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

Proving VP≠VNP a b -c d c d Affine map L: Mn(F)  Mk(F) is good if Pern = Detk L k(n): the smallest k for which there is a good map [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 c d

Proving polynomial identities

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 efficient probabilistic 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 efficient deterministic algorithm? [Kabanets-Impagliazzo] efficient deterministic algorithm (for proving algebraic identities)  “VPVNP” Proof: Pseudorandomness, Diagonalization

How to solve PIT? [KI’01] efficient deterministic PIT algorithm  “VPVNP” Use assumptions/Change the problem/Weaken the problem [KI’01] VPVNP  efficient deterministic 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 Kabanets-Impagliazzo Dvir-Shpilka Kayal-Saxena Kayal-Saraf

[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)

PIT: symbolic matrices singularity 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 L31 L32 L33 L21 L22 L23 L11 L12 L13 What we do solve: xi do not commute in F<(x1,x2,…,xm)> (free skew field) [Cohn’75] NC-SING decidable [CR’99] NC-SING  EXP [GGOW’15] NC-SING  P (F=Q) [IQS’16] NC-SING P (F large)

Non-commutative algebra

Non-commutative identities Word problem for free skew fields 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 not pq-1 or p-1q Nested inversion = (x + y)-1 - x-1 Hua’s identity

Non-commutative algebra Word problem for free skew fields 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! Generalizing [Valiant]

Invariant Theory symmetries, group actions, orbits, invariants - Energy - Momentum 1 2 3 4 5 S5 1 2 3 4 5 ??? Area

Invariant theory Linear actions 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=Sn acts on V=Fn by permuting coordinates VG = < elementary symmetric polynomials > Ex2: G=SLn(F)2 acts 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)2 acts on V by simultaneous basis changes R,C A RAC =(RA1C,RA2C,…,RAmC) Invariant ring Generators: few, low degree, easily computable

Invariant theory Left-Right action A=(A1,A2,…,Am)  Mn(F)m =V G=SLn(F)2 acts on V: RAC =(RA1C,RA2C,…,RAmC) (Zi)jk mn2 (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 ?  A NC-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! In P Derksen-Weyman, Domokos-Zubkov, Schofield-Van-der-berg

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)