In this paper we study algebraic branching programs (ABPs) with restrictions on the order and the number of reads of variables in the program. An ABP is given by a layered directed acyclic graph with source s and sink t, whose edges are labeled by variables taken from the set {x1, x2, . . . , xn} or field constants. It computes the sum of weights of all paths from s to t, where the weight of a ...

A polynomial identity testing algorithm must determine whether a given input polynomial is identically equal to 0. We give a deterministic black-box identity testing algorithm for univariate polynomials of the form $\sum_{j=0}^t c_j X^{\alpha_j} (a + b X)^{\beta_j}$. From our algorithm we derive an exponential lower bound for representations of polynomials such as $\prod_{i=1}^{2^n} (X^i-1)$ un...

A certain type of integer grid, called here an echelon grid, is an object found both in coherent systems whose components have a finite or countable number of levels and in algebraic geometry. If α= (α1, . . . , αd) is an integer vector representing the state of a system, then the corresponding algebraic object is a monomial x1 1 · · ·x αd d in the indeterminates x1, . . . , xd. The idea is to ...

We observe that a certain kind of algebraic proof—which covers essentially all known algebraic circuit lower bounds to date—cannot be used to prove lower bounds against VP if and only if what we call succinct hitting sets exist for VP. This is analogous to the Razborov–Rudich natural proofs barrier in Boolean circuit complexity, in that we rule out a large class of lower bound techniques under ...

We study the problem of obtaining efficient, deterministic, black-box polynomial identity testing algorithms (PIT) for algebraic branching programs (ABPs) that are read-once and oblivious. This class has an efficient, deterministic, white-box polynomial identity testing algorithm (due to Raz and Shpilka [RS05]), but prior to this work there was no known such black-box algorithm. The main result...

In this paper we study polynomial identity testing of sums of k read-once algebraic branching programs (Σk-RO-ABPs), generalizing the work of Shpilka and Volkovich [1, 2], who considered sums of k read-once formulas (Σk-RO-formulas). We show that Σk-RO-ABPs are strictly more powerful than Σk-RO-formulas, for any k ≤ ⌊n/2⌋, where n is the number of variables. Nevertheless, as a starting observat...