Algebraic independence and blackbox identity testing

Jacobian hits circuits

“…Thus, if each variable occurs in at most n 1/2−ε (0 < ε < 1/2) many underlying sparse polynomials, it takes an exponential sized depth-4 circuit to compute Imm n. Our next result is an exponential lower bound on the model for which hittingset was developed in [BMS11a] (but no lower bound was shown). It is also an improvement over the result obtained in [GKPS11] which holds only for more restricted depth-4 circuits over reals.…”

“…Our contribution, in this section, is an elementary proof of Theorem 2.3, which was originally proved in [BMS11a] using Krull's Hauptidealsatz. Here, we state the main properties of the Jacobian and faithful homomorphisms without proofs -for details, refer to [BMS11b] (or Appendix A.1).…”

“…Here, we state the main properties of the Jacobian and faithful homomorphisms without proofs -for details, refer to [BMS11b] (or Appendix A.1).…”

“…With depth-4 as the final frontier, the results that have been achieved so far include polynomial time hitting-set generators for the following models: • depth-2 (ΣΠ) circuits (equivalently, the class of sparse polynomials) [KS01], • depth-3 (ΣΠΣ) circuits with constant top fanin [SS11], • depth-4 (ΣΠΣΠ) multilinear circuits with constant top fanin [SV11], • constant-depth constant-read multilinear formulas [AvMV11] (& sparse-substituted variants), • circuits generated by sparse polynomials with constant transcendence degree [BMS11a]. To our knowledge, these are the only instances for which polynomial time hitting-set generators are known.…”

“…At a high level, this involved a study of the structure of multilinear formulas under the application of partial derivatives with respect to a carefully chosen set of variables and invoking depth-3 rank bounds (survey [SY10]). More recently, a third technique has emerged in [BMS11a] which is based upon the abstract concept of algebraic independence of polynomials and a related computational handle called the Jacobian. They showed that for any given poly-degree circuit C and sparse polynomials f 1, .…”