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We consider the cyclotomic identity testing (CIT) problem: given a polynomial $f(x_1,\ldots,x_k)$, decide whether $f(\zeta_n^{e_1},\ldots,\zeta_n^{e_k})$ is zero, where $\zeta_n = e^{2\pi i/n}$ is a primitive complex $n$-th root of unity…

Let $\Delta$ be a numerical semigroup and let $d\ge 2$ be an integer. We study the fiber of the quotient map \(S\mapsto S/d\) over $\Delta$. We describe its elements as semigroups of the form $\langle X\rangle+d\Delta$, for suitable finite…

We detail for the first time a complete explicit description of the quasi-cyclic structure of all classical finite generalized quadrangles. Using these descriptions we construct families of quasi-cyclic LDPC codes derived from the…

We study the computational complexity of decomposing finite discrete dynamical systems (FDDSs) in terms of the semiring operations of alternative and synchronous execution, which is useful for the analysis of discrete phenomena in science…

Let $\mathrm{R}$ be a real closed field. We prove that for any fixed $d$, the equivariant rational cohomology groups of closed symmetric semi-algebraic subsets of $\mathrm{R}^k$ defined by polynomials of degrees bounded by $d$ vanishes in…

Linear quasi-cyclic product codes over finite fields are investigated. Given the generating set in the form of a reduced Gr{\"o}bner basis of a quasi-cyclic component code and the generator polynomial of a second cyclic component code, an…

Let $\{a_1,\dots,a_p\}$ be the minimal generating set of a numerical monoid $S$. For any $s\in S$, its Delta set is defined by $\Delta(s)=\{l_{i}-l_{i-1}|i=2,\dots,k\}$ where $\{l_1<\dots<l_k\}$ is the set $\{\sum_{i=1}^px_i\,|\,…

We consider the problem of finding a small hitting set in an {\it infinite} range space $\cF=(Q,\cR)$ of bounded VC-dimension. We show that, under reasonably general assumptions, the infinite dimensional convex relaxation can be solved…

In this paper, we study the algebraic formula complexity of multiplying $d$ many $2\times 2$ matrices, denoted $\mathrm{IMM}_{d}$, and show that the well-known divide-and-conquer algorithm cannot be significantly improved at any depth, as…

Let $\delta(\Pc) = (\delta_0, \delta_1,..., \delta_d)$ be the $\delta$-vector of an integral polytope $\Pc \subset \RR^N$ of dimension $d$. Following the previous work of characterizing the $\delta$-vectors with $\sum_{i=0}^d \delta_i \leq…

We study the class of non-commutative Unambiguous circuits or Unique-Parse-Tree (UPT) circuits, and a related model of Few-Parse-Trees (FewPT) circuits (which were recently introduced by Lagarde, Malod and Perifel [LMP16] and Lagarde,…

This work extends the results known for the Delta sets of non-symmetric numerical semigroups with embedding dimension three to the symmetric case. Thus, we have a fast algorithm to compute the Delta set of any embedding dimension three…

A polynomial identity testing algorithm must determine whether an input polynomial (given for instance by an arithmetic circuit) is identically equal to 0. In this paper, we show that a deterministic black-box identity testing algorithm for…

We show that the rank of a depth-3 circuit (over any field) that is simple, minimal and zero is at most k^3\log d. The previous best rank bound known was 2^{O(k^2)}(\log d)^{k-2} by Dvir and Shpilka (STOC 2005). This almost resolves the…

In recent years, there has been a flurry of activity towards proving lower bounds for homogeneous depth-4 arithmetic circuits, which has brought us very close to statements that are known to imply $\textsf{VP} \neq \textsf{VNP}$. It is open…

Let $R$ be a real closed field. We consider basic semi-algebraic sets defined by $n$-variate equations/inequalities of $s$ symmetric polynomials and an equivariant family of polynomials, all of them of degree bounded by $2d < n$. Such a…

With the aim of derive a quasi-monomiality formulation in the context of discrete hypercomplex variables, one will amalgamate through a Clifford-algebraic structure of signature $(0,n)$ the umbral calculus framework with Lie-algebraic…

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.…

In this article we mainly consider the positively Z-graded polynomial ring R=F[X,Y] over an arbitrary field F and Hilbert series of finitely generated graded R-modules. The central result is an arithmetic criterion for such a series to be…

We study sparse polynomials with bounded individual degree and their factors, obtaining the following structural and algorithmic results. 1. A deterministic polynomial-time algorithm to find all sparse divisors of a sparse polynomial of…