Hitting Sets for Orbits of Circuit Classes and Polynomial Families

Abstract

The orbit of an n -variate polynomial \(f({\mathbf {x}})\) over a field \(\mathbb {F}\) is the set \(\text {orb}(f) := \lbrace f(A{\mathbf {x}}+{\mathbf {b}}) : A \in \mathrm{GL}(n,\mathbb {F}) \text{ and } {\mathbf {b}}\in \mathbb {F}^n\rbrace\) . The orbit of a polynomial f is a geometrically interesting subset of the set of affine projections of f . Affine projections of polynomials computable by seemingly weak circuit classes can be quite powerful. For example, the polynomial \(\mathsf {IMM}_{3,d}\) —the \((1,1)\) th entry of a product of d generic \(3 \times 3\) matrices—is computable by a constant-width read-once oblivious algebraic branching program (ROABP), yet every polynomial computable by a size- s general arithmetic formula is an affine projection of \(\mathsf {IMM}_{3,\text {poly}(s)}\) as shown by Ben-or and Cleve [ 12 ]. To our knowledge, no efficient hitting set construction was known for \(\text {orb}(\mathsf {IMM}_{3, d})\) before this work. In this article, we initiate the study of explicit hitting sets for the orbits of polynomials computable by several natural and well-studied circuit classes and polynomial families. In particular, we give quasi-polynomial time hitting sets for the orbits of: Low-individual-degree polynomials computable by commutative ROABPs . This implies quasi-polynomial time hitting sets for the orbits of the elementary symmetric polynomials and the orbits of multilinear sparse polynomials . Multilinear polynomials computable by constant-width ROABPs . This implies a quasi-polynomial time hitting set for the orbits of the family \(\lbrace \mathsf {IMM}_{3,d}\rbrace _{d \in \mathbb {N}}\) . Polynomials computable by constant-depth, constant-occur formulas . This implies quasi-polynomial time hitting sets for the orbits of multilinear depth-4 circuits with constant top fan-in , and also polynomial-time hitting sets for the orbits of the power symmetric polynomials and the sum-product polynomials . Polynomials computable by occur-once formulas . We say a polynomial has low individual degree if the degree of every variable in the polynomial is at most \(\text {poly}(\log n)\) , where n is the number of variables. The first two results are obtained by building upon and strengthening the rank concentration by translation technique of Agrawal, Saha, and Saxena [ 6 ]; the second result additionally uses the merge-and-reduce idea from Forbes and Shpilka [ 30 ] and Forbes, Shpilka, and Saptharishi [ 27 ]. The proof of the third result adapts the algebraic independence-based technique of Agrawal, Saha, Saptharishi, and Saxena [ 5 ] and Beecken, Mittmann, and Saxena [ 11 ] to reduce to the case of constructing hitting sets for the orbits of sparse polynomials. A similar reduction using the Shpilka-Volkovich (SV) generator-based argument in Shpilka and Volkovich [ 90 ] yields the fourth result. The SV generator plays an important role in all four results.