Finding irreducible polynomials over finite fields
STOC '86 Proceedings of the eighteenth annual ACM symposium on Theory of computing
Checking polynomial identities over any field: towards a derandomization?
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Fast Probabilistic Algorithms for Verification of Polynomial Identities
Journal of the ACM (JACM)
Reducing Randomness via Irrational Numbers
SIAM Journal on Computing
Probabilistic algorithms for sparse polynomials
EUROSAM '79 Proceedings of the International Symposiumon on Symbolic and Algebraic Computation
Primality and identity testing via Chinese remaindering
Journal of the ACM (JACM)
Testing polynomials which are easy to compute (Extended Abstract)
STOC '80 Proceedings of the twelfth annual ACM symposium on Theory of computing
Derandomizing polynomial identity tests means proving circuit lower bounds
Computational Complexity
CCC '08 Proceedings of the 2008 IEEE 23rd Annual Conference on Computational Complexity
Arithmetic Circuits: A Chasm at Depth Four
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Extractors And Rank Extractors For Polynomial Sources
Computational Complexity
The Complexity of the Annihilating Polynomial
CCC '09 Proceedings of the 2009 24th Annual IEEE Conference on Computational Complexity
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Arithmetic Circuits: A survey of recent results and open questions
Foundations and Trends® in Theoretical Computer Science
Black-box identity testing of depth-4 multilinear circuits
Proceedings of the forty-third annual ACM symposium on Theory of computing
Blackbox identity testing for bounded top fanin depth-3 circuits: the field doesn't matter
Proceedings of the forty-third annual ACM symposium on Theory of computing
Derandomizing Polynomial Identity Testing for Multilinear Constant-Read Formulae
CCC '11 Proceedings of the 2011 IEEE 26th Annual Conference on Computational Complexity
An Almost Optimal Rank Bound for Depth-3 Identities
SIAM Journal on Computing
Proving lower bounds via pseudo-random generators
FSTTCS '05 Proceedings of the 25th international conference on Foundations of Software Technology and Theoretical Computer Science
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
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Algebraic independence is an advanced notion in commutative algebra that generalizes independence of linear polynomials to higher degree. The transcendence degree (trdeg) of a set {f1, ..., fm} ⊂ F[x1, ..., xn] of polynomials is the maximal size r of an algebraically independent subset. In this paper we design blackbox and efficient linear maps ϕ that reduce the number of variables from n to r but maintain trdeg{ϕ(fi)}i = r, assuming fi's sparse and small r. We apply these fundamental maps to solve several cases of blackbox identity testing: 1. Given a circuit C and sparse subcircuits f1, ..., fm of trdeg r such that D := C(f1, ..., fm) has polynomial degree, we can test blackbox D for zeroness in poly(size(D))r time. 2. Define a ΣΠΣΠδ(k, s, n) circuit C to be of the form Σi=1k Πj=1s fi,j, where fi,j are sparse n-variate polynomials of degree at most δ. For k = 2, we give a poly (δsn)δ2 time blackbox identity test. 3. For a general depth-4 circuit we define a notion of rank. Assuming there is a rank bound R for minimal simple ΣΠΣΠδ(k, s, n) identities, we give a poly(δsnR)Rkδ2 time blackbox identity test for ΣΠΣΠδ(k, s, n) circuits. This partially generalizes the state of the art of depth-3 to depth-4 circuits. The notion of trdeg works best with large or zero characteristic, but we also give versions of our results for arbitrary fields.