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 = ∏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’’ ?

4. Arithmetic Complexity

5. 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)

6. 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

7. 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?

8. 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

9. 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

10. 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|

11. 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

12. 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 Per = 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

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 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

15. 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

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. 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)

18. Non-commutative algebra

19. 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

20. 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]

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

22. 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

23. 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

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)