Randomized Black-Box PIT for Small Depth +-Regular Non-commutative Circuits

TL;DR

This work resolves the black-box PIT problem for non-commutative polynomials computed by constant-depth $+$-regular circuits. The authors develop a novel pipeline that converts depth-$d$ circuits into structured ordered power-sum representations via substitution automata, sparsifies products, applies commutative reductions, and modifies coefficients, ultimately enabling a single matrix-substitution PIT test. They prove that a non-zero polynomial $f$ of degree $D$ computed by such a circuit of size $s$ cannot vanish on the matrix algebra $\mathbb{M}_{N}(\mathbb{F})$ with $N=s^{O(d^2)}$ (and suitable field size depending on $D$), yielding a randomized polynomial-time PIT algorithm for constant depth. The key novelty lies in the matrix-composition technique that aggregates multiple substitution steps into one, together with the coefficient-modification mechanism that prevents cancellation. The results substantially advance PIT in the non-commutative setting and open paths toward broader circuit classes, with practical implications for algebraic complexity and identity testing under black-box access.

Abstract

We address the black-box polynomial identity testing (PIT) problem for non-commutative polynomials computed by $+$-regular circuits, a class of homogeneous circuits introduced by [AJMR](STOC 2017, Theory of Computing 2019). These circuits can compute polynomials with a number of monomials that are doubly exponential in the circuit size. They gave an efficient randomized PIT algorithm for +-regular circuits of depth 3 and posed the problem of developing an efficient black-box PIT for higher depths as an open problem. We present a randomized black-box polynomial-time algorithm for +-regular circuits of any constant depth. Specifically, our algorithm runs in $s^{O(d^2)}$ time, where $s$ and $d$ represent the size and the depth of the $+$-regular circuit, respectively. We combine several key techniques in a novel way. We employ a nondeterministic substitution automaton that transforms the polynomial into a structured form and utilizes polynomial sparsification along with commutative transformations to maintain non-zeroness. Additionally, we introduce matrix composition, coefficient modification via the automaton, and multi-entry outputs-methods that have not previously been applied in the context of black-box PIT. Together, these techniques enable us to effectively handle exponential degrees and doubly exponential sparsity in non-commutative settings, enabling polynomial identity testing for higher-depth circuits. Our work resolves an open problem from [AJMR]. In particular, we show that if $f$ is a non-zero non-commutative polynomial in $n$ variables over the field $\mathbb{F}$, computed by a depth-$d$ $+$-regular circuit of size $s$, then $f$ cannot be a polynomial identity for the matrix algebra $\mathbb{M}_{N}(\mathbb{F})$, where $N= s^{O(d^2)}$ and the size of the field $\mathbb{F}$ depending on the degree of $f$.

Figures (6)

• Figure 1: The transition diagram for the variable $x_i : 1\leq i\leq n$
• Figure 2: Substitution automaton for product sparsification
• Figure 3: Automaton that converts n.c. variables to commutative variables
• Figure 4: Substitution automaton that computes number of $\xi$-patterns modulo $p$.
• Figure 5: Modulo counting automaton that changes coefficients. Here, $\lambda\equiv D_1 \mod p$

Theorems & Definitions (62)

• Definition 1.0.1: Non-commutative arithmetic circuits
• Definition 1.0.2: $+$-regular circuits AJMR19
• Remark 1.0.1
• Theorem 1
• Definition 1.1.1: Boundary of a Polynomial
• Definition 2.1.1: Substitution NFA
• Definition 2.2.1
• Lemma 2.2.1
• Lemma 2.3.1: DeMillo-Lipton-Schwartz-Zippel Lemma DL78Zippel79Sch80
• Lemma 3.0.1: Composing $\mathrm{K}$ Substitution Matrices