Constructing $r$-th nonresidue over a finite field is a fundamental computational problem. A related problem is to construct an irreducible polynomial of degree $r^e$ (where $r$ is a prime) over a given finite field $\F_q$ of characteristic $p$ (equivalently, constructing the bigger field $\F_{q^{r^e}}$). Both these problems have famous randomized ... more >>>

We design polynomial size, constant depth (namely, $AC^0$) arithmetic formulae for the greatest common divisor (GCD) of two polynomials, as well as the related problems of the discriminant, resultant, Bézout coefficients, squarefree decomposition, and the inversion of structured matrices like Sylvester and Bézout matrices. Our GCD algorithm extends to any ... more >>>

We show that the GCD of two univariate polynomials can be computed by (piece-wise) algebraic circuits of constant depth and polynomial size over any sufficiently large field, regardless of the characteristic. This extends a recent result of Andrews \& Wigderson who showed such an upper bound over fields of zero ... more >>>

We show that Hilbert's Nullstellensatz, the problem of deciding if a system of multivariate polynomial equations has a solution in the algebraic closure of the underlying field, lies in the counting hierarchy. More generally, we show that the number of solutions to a system of equations can be computed in ... more >>>

The problem of decomposing an algebraic variety into more structured components (such as equidimensional, irreducible, or connected components) is a fundamental problem in computer algebra and algebraic geometry. In this work, we study the computational complexity of computing equidimensional components of algebraic varieties, represented via Chow forms.

We improve the ... more >>>

Andrews, Garg, and Schost (FOCS~2026) recently, in a breakthrough result, placed Hilbert's Nullstellensatz (HN) in the Counting Hierarchy ($CH$) using constant-depth circuits for multivariate resultants. While HN deals with exact polynomial satisfiability, Guo, Saxena, and Sinhababu (CCC~2018; Theory of Computing~2019) introduced approximate polynomial satisfiability (APS) as a natural approximative analogue ... more >>>