Related papers: Identity Testing for Radical Expressions

We study the problem of polynomial identity testing (PIT) for depth 2 arithmetic circuits over matrix algebra. We show that identity testing of depth 3 (Sigma-Pi-Sigma) arithmetic circuits over a field F is polynomial time equivalent to…

We review the history and previous literature on radical equations and present the rigorous solution theory for radical equations of depth 2, continuing a previous study of radical equations of depth 1. Radical equations of depth 2 are…

$ \newcommand{\ie}{i.\,e.} $We introduce a hitting set generator for Polynomial Identity Testing based on evaluations of low-degree univariate rational functions at abscissas associated with the variables. We establish an equivalence up to…

In the classical linear degeneracy testing problem, we are given $n$ real numbers and a $k$-variate linear polynomial $F$, for some constant $k$, and have to determine whether there exist $k$ numbers $a_1,\ldots,a_k$ from the set such that…

Motivated by the quantum algorithm in \cite{MN05} for testing commutativity of black-box groups, we study the following problem: Given a black-box finite ring $R=\angle{r_1,...,r_k}$ where $\{r_1,r_2,...,r_k\}$ is an additive generating set…

We study arithmetic proof systems P_c(F) and P_f(F) operating with arithmetic circuits and arithmetic formulas, respectively, that prove polynomial identities over a field F. We establish a series of structural theorems about these proof…

In this paper we show that the black-box polynomial identity testing for noncommutative polynomials $f\in\mathbb{F}\langle z_1,z_2,\cdots,z_n \rangle$ of degree $D$ and sparsity $t$, can be done in randomized $\poly(n,\log t,\log D)$ time.…

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…

We deploy numerical semidefinite programming and conversion to exact rational inequalities to certify that for a positive semidefinite input polynomial or rational function, any representation as a fraction of sums-of-squares of polynomials…

Powerful skew arithmetic circuits are introduced. These are skew arithmetic circuits with variables, where input gates can be labelled with powers $x^n$ for binary encoded numbers $n$. It is shown that polynomial identity testing for…

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.…

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…

This paper investigates $\exists\mathbb{R}(r^{\mathbb{Z}})$, that is the extension of the existential theory of the reals by an additional unary predicate $r^{\mathbb{Z}}$ for the integer powers of a fixed computable real number $r > 0$. If…

Motivated by practical applications in the design of optimization compilers for neural networks, we initiated the study of identity testing problems for arithmetic circuits augmented with \emph{exponentiation gates} that compute the real…

Motivated by the Hadamard product of matrices we define the Hadamard product of multivariate polynomials and study its arithmetic circuit and branching program complexity. We also give applications and connections to polynomial identity…

Let $\mathbf{f}=(f\_1,\ldots,f\_m)$ and $\mathbf{g}=(g\_1,\ldots,g\_m)$ be two sets of $m\geq 1$ nonlinear polynomials over $\mathbb{K}[x\_1,\ldots,x\_n]$ ($\mathbb{K}$ being a field). We consider the computational problem of finding -- if…

Radical membership testing, and the special case of Hilbert's Nullstellensatz (HN), is a fundamental computational algebra problem. It is NP-hard; and has a famous PSPACE algorithm due to effective Nullstellensatz bounds. We identify a…

The circuit evaluation problem (also known as the compressed word problem) for finitely generated linear groups is studied. The best upper bound for this problem is $\mathsf{coRP}$, which is shown by a reduction to polynomial identity…

This paper is concerned with certifying that a given point is near an exact root of an overdetermined or singular polynomial system with rational coefficients. The difficulty lies in the fact that consistency of overdetermined systems is…

The radical solution of polynomials with rational coefficients is a famous solved problem. This paper found that it is a $\mathbb{NP}$ problem. Furthermore, this paper found that arbitrary $ \mathscr{P} \in \mathbb{P}$ shall have a one-way…