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Explore the complexity of algebraic identities, word problems, arithmetic computations, and symbolic matrices in the realm of mathematics and computation, with a focus on verifying identities efficiently. Learn about non-commutative singularity, NC algebra, and invariant theory in this comprehensive study. Delve into the intricate world of symbolic determinants, polynomial representations, and the profound connection to computational complexity. Discover the significance of polynomial identity testing (PIT) and the implications for the VP vs. VNP problem.
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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 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)
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)
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
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?
VP = VNP ? [Valiant’79] char(F)2. XMn(F) matrix of variables xij Detn(X) = Snsgn() i[n] xi(i) Pern(X) = Sni[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
[Gauss+]: DetVP: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
[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|
VP≠VNP? L11 L12 L13 L21 L22 L23 Does VPVNP? Does Per have small formula? Does Per have small symbolic determinant? Pern(X) = Detk( )Lij(X) affine, ksmall? L31 L32 L33
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) VPVNP [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
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) “VPVNP” Proof: Pseudorandomness, Diagonalization
How to solve PIT? [KI’01] efficientdeterministic PIT algorithm “VPVNP” Use assumptions/Change the problem/Weaken the problem [KI’01] VPVNP 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
[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)
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] SINGRP 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-SINGEXP [GGOW’15] NC-SING P (F=Q) [IQS’16] NC-SINGP (F large)
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
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 AiDi) = 0 d, Di Md(F) Det(i AiXi) = 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]
R Invariant Theorysymmetries, group actions,orbits, invariants - Energy - Momentum 1 1 5 2 5 2 S5 4 3 4 3 ??? Area
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
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 ZiDi) : d N, Di Md(F)> Nullcone: Given A, does p(A)=0 for all pF[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
Summary • Algebraic identities & word problems • Symbolic determinant & inverse: completeness • NC-SING: solved (Commutative Invariant Th., Quantum Inf. Th.) • C-SING:open (de-randomization,VPVNP) • Is C-SING a group invariant? • New efficient algorithms in Invariant Theory (nullcone, orbit closure intersection, other actions)