Related papers: Integer Complexity Generalizations in Various Ring…

We study the regularity and the projective dimension of the Stanley-Reisner ring of a $k$-decomposable simplicial complex and explain these invariants with a recursive formula. To this aim, the graded Betti numbers of $k$-decomposable…

It is known that point searching in basic semialgebraic sets and the search for globally minimal points in polynomial optimization tasks can be carried out using $(s\,d)^{O(n)}$ arithmetic operations, where $n$ and $s$ are the numbers of…

We generalize a result of Eisenbud-Huneke-Ulrich on the maximal graded shifts of a module with prescribed annihilator and prove a linear regularity bound for ideals in a polynomial ring depending only on the first $p - c$ steps in the…

Recently, metric learning and similarity learning have attracted a large amount of interest. Many models and optimisation algorithms have been proposed. However, there is relatively little work on the generalization analysis of such…

We prove that any mixed-integer linear extended formulation for the matching polytope of the complete graph on $n$ vertices, with a polynomial number of constraints, requires $\Omega(\sqrt{\sfrac{n}{\log n}})$ many integer variables. By…

We investigate degree bounds for fields of rational invariants of representations of finite groups. We prove many cases of a bound for $\mathbb{Z}/p\mathbb{Z}$ conjectured by Blum-Smith, Garcia, Hidalgo, and Rodriguez. For arbitrary groups,…

We consider the problem of computing succinct encodings of lists of generators for invariant rings for group actions. Mulmuley conjectured that there are always polynomial sized such encodings for invariant rings of…

We develop the notions of Newton non-degenerate (NND) ideals and Newton polyhedra for regular local rings. These concepts were first defined in the context of complex analysis. We show that the characterization of NND ideals via their…

Our results are of three types. First we describe a general procedure of adjoining polynomial variables to $A_\infty$-ring spectra whose coefficient rings satisfy certain restrictions.A host of examples of such spectra is provided by…

The average density of zeros for monic generalized polynomials, $P_n(z)=\phi(z)+\sum_{k=1}^nc_kf_k(z)$, with real holomorphic $\phi ,f_k$ and real Gaussian coefficients is expressed in terms of correlation functions of the values of the…

Different and distinct notions of regularity for modules exist in the literature. When these notions are restricted to commutative rings, they all coincide with the well-known von-Neumann regularity for rings. We give new characterizations…

In this article, We introduce a condition that is both necessary and sufficient for a linear code to achieve minimality when analyzed over the rings $\mathbb{Z}_{n}$.The fundamental inquiry in minimal linear codes is the existence of a…

We define two new families of polynomials that generalize permanents and prove upper and lower bounds on their determinantal complexities comparable to the known bounds for permanents. One of these families is obtained by replacing…

Define $||n||$ to be the complexity of $n$, the smallest number of ones needed to write $n$ using an arbitrary combination of addition and multiplication. The set $\mathscr{D}$ of defects, differences $\delta(n):=||n||-3\log_3 n$, is known…

We present a uniform description of sets of $m$ linear forms in $n$ variables over the field of rational numbers whose computation requires $m(n - 1)$ additions.

Let M be a finitely generated ZZ-graded module over the standard graded polynomial ring R=K[X_1, ..., X_n] with K a field, and let H_M(t)=Q_M(t)/(1-t)^d be the Hilbert series of M. We introduce the Hilbert regularity of M as the lowest…

We classify all cyclotomic matrices over real quadratic integer rings and we show that this classification is the same as classifying cyclotomic matrices over the compositum all real quadratic integer rings. Moreover, we enumerate a related…

The existence of string functions, which are not polynomial time computable, but whose graph is checkable in polynomial time, is a basic assumption in cryptography. We prove that in the framework of algebraic complexity, there are no such…

We consider the problem of computing matrix polynomials $p(X)$, where $X$ is a large dense matrix, with as few matrix-matrix multiplications as possible. More precisely, let $\Pi_{2^{m}}^*$ represent the set of polynomials computable with…

We determine the exact values of the linear complexity of 2p-periodic quaternary sequences over Z_4 (the residue class ring modulo 4) defined from the generalized cyclotomic classes modulo 2p in terms of the theory of of Galois rings of…