Algebraic Independence and Blackbox Identity Testing

Abstract

Abstract. 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, . (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.

Progress on Polynomial Identity Testing-II

“…There are several notions of rank in commutative algebra. The one we [BMS13] found useful istranscendence degree (trdeg). We say that a set S of polynomials {f 1 , .…”

“…With this notion we call a homomorphism ϕ faithful if trdeg(S) = trdeg(ϕ(S)). The usefulness of ϕ (assuming that one can come up with it efficiently) was first proved in [BMS13]: This implies that we can use a faithful map to 'reduce' the number of variables n without changing the nonzeroness of C. The strategy can be used in cases where trdeg(f ) is small, say, smaller than a constant r.…”

“…To do this we need three fundamental ingredients -an efficient criterion for algebraic independence (Jacobian), its behavior under ϕ (chain rule), and standard maps (Vandermonde & Kronecker based). There are several proofs of this, see [Jac41,For91,BMS13,MSS12]. This gives us an efficient way to capture trdeg, when the characteristic is zero/large.…”

From sylvester-gallai configurations to rank bounds

“…As for blackbox algorithms, the authors are quite sure that the reader has heard enough history. Identity tests are known only for special depth-4 circuits [Arvind and Mukhopadhyay 2010;Saxena 2008;Shpilka andVolkovich 2008, 2009;Karnin et al 2010;Saraf and Volkovich 2011;Anderson et al 2011;Beecken et al 2011;Saha et al 2011]. Recently, Agrawal et al [2011] presents a unified approach to study diverse circuit restrictions, by generalizing the notion of rank and employing Jacobian techniques.…”

Additive Combinatorics over Finite Fields: New Results and Applications

“…Certainly these problems are special cases of the more general problems of oracle (also sometimes called "black-box") polynomial interpolation and identity testing for arbitrary polynomials, see [13] and references therein.…”