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Modules are a generalization of the vector spaces of linear algebra over a ring instead of over a field of scalars. Some of the results in linear algebra are extremely useful in studying modules. In this paper, we highlight the use of Smith normal form in studying finitely generated modules over a univariate polynomial ring. As an application, we take advantage of Smith normal form to understand the structure of certain syzygy modules. [ABSTRACT FROM AUTHOR]

Let G = Cp × H be a finite group, where Cp is a cyclic group of prime order p and H is a p'-group. Let F be an algebraically closed field in characteristic p. Let V be a direct sum of m non-trivial indecomposable G-modules such that the norm polynomials of the simple H-modules are the power of the product of the basis elements of the dual. In previous work, we proved the periodicity property of the polynomial ring F[V] with actions of G. In this paper, by the periodicity property, we showed that F[V]G is generated by m norm polynomials together with homogeneous invariants of degree at most m|G| - dim(V) and transfer invariants, which yields the well-known degree bound dim(V)·(|G|-1). More precisely, we found that this bound gets less sharp as the dimensions of simple H-modules increase. [ABSTRACT FROM AUTHOR]

The interactions between topology and algebraic geometry expose various interesting properties. This paper proposes the deformations of topological n-manifolds over the automorphic polynomial ring maps and associated isomorphic imbedding of locally flat submanifolds within the n-manifolds. The manifold deformations include topologically homeomorphic bending of submanifolds at multiple directions under algebraic operations. This paper introduces the concept of a topological equivalence class of manifolds and the associated equivalent class of polynomials in a real ring. The concepts of algebraic compositions in a real polynomial ring and the resulting topological properties (homeomorphism, isomorphism and deformation) of manifolds under algebraic compositions are introduced. It is shown that a set of ideals in a polynomial ring generates manifolds retaining topological isomorphism under algebraic compositions. The numerical simulations are presented in this paper to illustrate the interplay of topological properties and the respective real algebraic sets generated by polynomials in a ring within affine 3-spaces. It is shown that the coefficients of polynomials generated by a periodic smooth function can induce mirror symmetry in manifolds. The proposed formulations do not consider the simplectic class of manifolds and associated quantizable deformations. However, the proposed formulations preserve the properties of Nash representations of real algebraic manifolds including Nash isomorphism. [ABSTRACT FROM AUTHOR]

Given a graph G whose edges are labeled by ideals of a commutative ring R with identity, a generalized spline is a vertex labeling of G by the elements of R so that the difference of labels on adjacent vertices is an element of the corresponding edge ideal. The set of all generalized splines on a graph G with base ring R has a ring and an R -module structure. In this paper, we focus on the freeness of generalized spline modules over certain graphs with the base ring R = k [ x 1 , ... , x d ] where k is a field. We first show the freeness of generalized spline modules on graphs with no interior edges over k [ x , y ] such as cycles or a disjoint union of cycles with free edges. Later, we consider graphs that can be decomposed into disjoint cycles without changing the isomorphism class of the syzygy modules. Then we use this decomposition to show that generalized spline modules are free over k [ x , y ] and later we extend this result to the base ring R = k [ x 1 , ... , x d ] under some restrictions. [ABSTRACT FROM AUTHOR]

An R-moduleM is called δ-weakly Hopfian if any δ-small surjective endomorphism of M is an automorphism. In this paper we explore various properties of δ-weakly Hopfian modules, shedding light on their distinct characteristics. Additionally, we examine the δ-weakly Hopficity of modules over polynomial and truncated polynomial rings. [ABSTRACT FROM AUTHOR]

Let K be a field and S = K[x1,..., xn] a standard polynomial ring over K. In this paper, we give new combinatorial algorithms to compute the smallest t-spread lexicographic set and the smallest t-spread strongly stable set containing a given set of t-spread monomials of S. Some technical tools allowing to compute the cardinality of t-spread strongly stable sets avoiding their construction are also presented. Such functions are also implemented in a Macaulay2 package, TSpreadIdeals, to ease the computation of well-known results about algebraic invariants for t-spread ideals. [ABSTRACT FROM AUTHOR]

In this paper, we use multivariate polynomial rings to construct quantum error-correcting codes (QECCs) via Hermitian construction. We establish a relation between linear codes and ideals of multivariate polynomial rings. We give a necessary and suffcient condition for a multivariate polynomial to generate a Hermitian dual-containing code. By comparing with the literatures in recent years, we construct 31 new QECCs over F q , where q = 3 , 4 , 5 , 7 . Some of them reach quantum singleton bound and some of them exceed quantum GV bound. [ABSTRACT FROM AUTHOR]

In this paper, we first consider the existence conditions, the construction and the enumeration of Hermitian self-dual θ -cyclic and θ -negacyclic codes over I F p 2 , where p is a prime number and θ is the Frobenius automorphism over I F p 2 . We then give necessary and sufficient conditions for the existence of Hermitian self-dual θ -cyclic and θ -negacyclic codes over I F p e where e is an even integer greater than 2. [ABSTRACT FROM AUTHOR]

In this paper, we study orbits and fixed points of polynomials in a general skew polynomial ring D [ x , σ , δ ]. We extend results of the first author and Vishkautsan on polynomial dynamics in D [ x ]. In particular, we show that if a ∈ D and f ∈ D [ x , σ , δ ] satisfy f (a) = a , then f ∘ n (a) = a for every formal power of f. More generally, we give a sufficient condition for a point a to be r -periodic with respect to a polynomial f. Our proofs build upon foundational results on skew polynomial rings due to Lam and Leroy. [ABSTRACT FROM AUTHOR]

In this paper, we prove a result similar to results of Itoh [J. Algebra 150 (1992), pp. 101–117] and Hong-Ulrich [J. Lond. Math. Soc. (2) 90 (2014), pp. 861–878], proving that integral closure of an ideal is compatible with specialization by a general element of that ideal for ideals of height at least two in a large class of rings. Moreover, we show integral closure of sufficiently large powers of the ideal is compatible with specialization by a general element of the original ideal. In a polynomial ring over an infinite field, we give a class of squarefree monomial ideals for which the integral closure is compatible with specialization by a general linear form. [ABSTRACT FROM AUTHOR]

Our paper delves into exploring skew cyclic codes over a generalized class of rings denoted by T = Tr. We define Tr = Fq[u1, u2, ..., ur]/(ui3 - ui, uiuj -ujui)i,jr=1, q = pm and p is some odd prime. Our study introduces a Gray map for the ring T and explores its properties. Using a decomposition theorem, we analyze the structural features of skew cyclic codes over T . Additionally, we offer a formula to find the count of skew cyclic codes of length n over the ring T under specific conditions. Further, we derive a criterion to get Linear Complementary Dual (LCD) codes over T from skew cyclic codes. Moreover, we present a technique for deriving quantum codes from a particular class of skew cyclic codes over T which contain their dual. [ABSTRACT FROM AUTHOR]

This paper presents the theoretical foundations of finding the inverse of a polynomial modulo in the ring Z[x] based on the method of undetermined coefficients. The use of the latter makes it possible to significantly reduce the time complexity of calculations avoiding the operation of finding the greatest common divisor. An example of calculating the inverse of a polynomial modulo in the ring Z [ x ] based on the proposed approach is given. Analytical expressions of the time complexities of the developed and classical methods depending on the degrees of polynomials are built. The graphic dependence of the complexity of performing the operation of finding the inverse of a polynomial in the ring Z[x] is presented, which shows the advantages of the method based on undetermined coefficients. It is found that the efficiency of the developed method increases logarithmically with an increase in the degrees of polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we strengthen a result by Green about an analogue of Sárközy's theorem in the setting of polynomial rings Fq[x]. In the integer setting, for a given polynomial F ∈ Z[x] with constant term zero, (a generalization of) Sárközy's theorem gives an upper bound on the maximum size of a subset A ⊂{1, ..., n} that does not contain distinct α1, α2 ∈ A satisfying α1, -- α2 = F(b) for some b ∈ Z. Green proved an analogous result with much stronger bounds in the setting of subsets A ⊂ Fq[x] of the polynomial ring Fq[x], but this result required the additional condition that the number of roots of the polynomial F ∈ Fq[x] be coprime to q. We generalize Green's result, removing this condition. As an application, we also obtain a version of Sárközy's theorem with similar strong bounds for subsets A ⊂ Fq for q=pn for a fixed prime p and large n. [ABSTRACT FROM AUTHOR]

Let K = Q(α) be a pure number field generated by a complex root α of a monic irreducible polynomial F(x) = x2r · 3k·7s− m ∈ Z[x], where r, k, s are three positive natural integers. The purpose of this paper is to study the monogenity of K. Our results are illustrated by some examples. [ABSTRACT FROM AUTHOR]

Let K = Q(a) be a pure number field generated by a root a of a monic irreducible polynomial xpr - a Z[x], where p is a rational prime and r is a positive integer. Let ZK be the ring of integers of K. In this paper, we calculate an integral basis of ZK and study the monogenity of K in some particular cases. [ABSTRACT FROM AUTHOR]

In a 1987 paper, Eliahou and Kervaire constructed a minimal resolution of a class of monomial ideals in a polynomial ring, called stable ideals. As a consequence of their construction they deduced several homological properties of stable ideals. Furthermore they showed that this resolution admits an associative, graded commutative product that satisfies the Leibniz rule. In this paper we show that their construction can be extended to stable ideals in skew polynomial rings. As a consequence we show that the homological properties of stable ideals proved by Eliahou and Kervaire hold also for stable ideals in skew polynomial rings. [ABSTRACT FROM AUTHOR]

In this paper, we classify the derivations of degree-one generalized Weyl algebras over a univariate Laurent polynomial ring. In particular, our results cover the Weyl–Hayashi algebra, a quantization of the first Weyl algebra arising as a primitive factor algebra of U q + ( 5) , and a family of algebras which localize to the group algebra of the infinite group with generators x and y , subject to the relation x y = y − 1 x. [ABSTRACT FROM AUTHOR]

Recall that a ring R is a Hilbert ring if any maximal ideal of R [ X ] contracts to a maximal ideal of R. The main purpose of this paper is to characterize the prime ideals of a commutative ring R which are traces of the maximal ideals of the polynomial ring R [ X ]. In this context, we prove that if p is a prime ideal of R such that R / p is a semi-local domain of (Krull) dimension ≤ 1 , then p is the trace of a maximal ideal of R [ X ]. Whereas, if R is Noetherian and either (dim (R / p) ≥ 2) or (the quotient field of R / p is algebraically closed, dim (R / p) = 1 and R / p is not semi-local), then p is never the trace of a maximal ideal of R [ X ]. Putting these results into use in investigating the Ext-index of Noetherian rings, we establish connections between the finiteness of the Ext-index of localizations of the polynomial rings R [ X ] and the finiteness of the Ext-index of localizations of R. This allows us to provide a new class of rings satisfying some known conjectures on Ext-index of Noetherian rings as well as to build bridges between these conjectures. [ABSTRACT FROM AUTHOR]

Let S be the polynomial ring on the space of non-square generic matrices or the space of odd-sized skew-symmetric matrices, and let I be the determinantal ideal of maximal minors or Pf the ideal of sub-maximal Pfaffians, respectively. Using desingularizations and representation theory of the general linear group we expand upon work of Raicu–Weyman–Witt [Adv. Math. 250 (2014), pp. 596–610] to determine the S-module structures of Ext^j_S(S/I^t, S) and Ext^j_S(S/Pf^t, S), from which we get the degrees of generators of these Ext modules. As a consequence, via graded local duality we answer a question of Wenliang Zhang [J. Pure Appl. Algebra 225 (2021), Paper No. 106789] on the socle degrees of local cohomology modules of the form H^j_\mathfrak {m}(S/I^t). [ABSTRACT FROM AUTHOR]

Sylvester-polynomial-conjugate matrix equations unify many well-known versions and generalizations of the Sylvester matrix equation A X − X B = C which have a wide range of applications. In this paper, we present a general approach to Sylvester-polynomial-conjugate matrix equations via groupoids, vector spaces, and matrices over skew polynomial rings. The obtained results are applied to Sylvester-polynomial-conjugate matrix equations over complex numbers and quaternions. The main role in our approach is played by skew polynomial rings, which are well-known tools in algebra to provide examples of asymmetry between left-sided and right-sided versions of many ring objects. [ABSTRACT FROM AUTHOR]

In the present paper, dedicated to Yuri Manin, we investigate the general notion of rings of S[μn, +​]-polynomials and relate this concept to the known notion of number systems. The Riemann–Roch theorem for the ring Z of the integers that we obtained recently uses the understanding of Z as a ring of polynomials S[X] in one variable over the absolute base S, where 1+1=X+X². The absolute base S (the categorical version of the sphere spectrum) thus turns out to be a strong candidate for the incarnation of the mysterious F1​. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the strong version of countably A-rings (or the countably McCoy rings) introduced by T. Lucas in [21]. Moreover, we introduce and investigate the module theoretic version of the strongly countably A-ring notion, namely the strongly countably A-modules. We focus especially on the behavior of the stronlgy countably A-property vis-à-vis the polynomial ring, the power series ring, the idealization and the amalgamated duplication of a ring along an ideal. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the structure and properties of skew negacyclic codes and skew quasi-negacyclic codes over the ring R = ℤ q + u ℤ q , u 2 = 0. Some structural properties of R [ x , Θ ] are discussed, where Θ is an automorphism of R. A skew quasi-negacyclic code of length s ℓ with index ℓ over R is viewed both as in the conventional row circulant form and also as an R [ x , Θ ] / (x s + 1) -submodule of G R (R , ℓ) [ x , δ ] / (x s + 1) , where G R (R , ℓ) is the Galois extension ring of degree ℓ over R and δ is an automorphism of G R (R , ℓ). A sufficient condition for one generator skew quasi-negacyclic codes to be free is determined. Some distance bounds for free one generator skew quasi-negacyclic codes are discussed. Furthermore, given the decomposition of a skew quasi-negacyclic code, we provide the decomposition of its dual code. As a result, a characterization of self-dual skew quasi-negacyclic codes over R = ℤ q + u ℤ q is provided. By using computer search we obtained a number of new linear codes over ℤ 4 from skew negacyclic and skew quasi-negacyclic codes over ℤ 4 + u ℤ 4 . [ABSTRACT FROM AUTHOR]

An equigenerated monomial ideal I in the polynomial ring S = K [ x 1 , ... , x n ] is a Freiman ideal if μ (I 2) = ℓ (I) μ (I) − ℓ (I) 2 , where ℓ (I) is the analytic spread of I and μ (I) is the least number of monomial generators of I. In this paper, we classify certain classes of Borel-type ideals, including Borel ideals with multiple Borel generators and principal k -Borel ideals, which are Freiman. [ABSTRACT FROM AUTHOR]

Let K be a field and K[x1, x2] the polynomial ring in two variables over K with each xi of degree 1. Let L be the generalized mixed product ideal induced by a monomial ideal I ⊂ K[x1, x2], where the ideals substituting the monomials in I are squarefree Veronese ideals. In this paper, we study the integral closure of L, and the normality of R(L), the Rees algebra of L. Furthermore, we give a geometric description of the integral closure of R(L). [ABSTRACT FROM AUTHOR]

In this paper, our main aim is to study periodic and potent elements of a ring. We specially study periodic elements of graded rings and generalize some classical results related to idempotent of polynomial rings. We show that a (von Neumann) quasi-inverse of a potent element is a root of unity. We study the isomorphism of potent elements and analyze some closure properties of the set Pot(R) of potent elements of a ring R. The potent elements of the endomorphism ring of a Fitting module are described, and we apply this to matrices over division rings. In the case of matrices over finite fields, we connect features of potent elements with the exponent of their minimal polynomials. [ABSTRACT FROM AUTHOR]

Principal symmetric ideals were recently introduced by Harada et al. in [The minimal free resolution of a general principal symmetric ideal, preprint (2023), arXiv:2308.03141], where their homological properties are elucidated. They are ideals generated by the orbit of a single polynomial under permutations of variables in a polynomial ring. In this paper, we determine when a product of two principal symmetric ideals is principal symmetric and when the powers of a principal symmetric ideal are again principal symmetric ideals. We characterize the ideals that have the latter property as being generated by polynomials invariant up to a scalar multiple under permutation of variables. Recognizing principal symmetric ideals is an open question for the purpose of which we produce certain obstructions. We also demonstrate that the Hilbert functions of symmetric monomial ideals are not all given by symmetric monomial ideals, in contrast to the non-symmetric case. [ABSTRACT FROM AUTHOR]

In this paper, we study the structure of the skew polynomial ring R = ( p + u p) [ x ; ] and its quotient ring R n = R / 〈 x n − 1 〉 , where p is an odd prime number, u 2 = 0 , and (u) = − u. We give an explicit structure of the ideals in R and R n and propose an algorithm to characterize them. We identify the structure of prime, maximal, and primary ideals in these rings. In particular, we prove that this group ring is not Laskerian. [ABSTRACT FROM AUTHOR]

We study the rings Nil D of nilpotent elements of R-derivations D on polynomial rings over a UFD R containing Q . The main result of this paper determines Nil D of the monomial derivations D on R [ x , y ] . Furthermore, as its application, we prove that, for each positive integer m, there exists a monomial R-derivation with a slice on R [ x , y , z ] such that the minimal number of generators of its kernel equals m. [ABSTRACT FROM AUTHOR]

The notion of c-differential uniformity was proposed by Ellingsen et al., which generalizes the classical differential uniformity measuring the resistance against differential cryptanalysis. Since then, the research of functions with low c-differential uniformity over finite fields attracted many researchers’ attention. However, it seems that there is no study of function with low c-differential uniformity over integer rings modulo n. In this paper, we give an extension of the c-differential uniformity concept to rings of integers module some n > 0, and we present several perfect c-nonlinear polynomial functions on the integer ring ℤn for the different integer n. [ABSTRACT FROM AUTHOR]

For a prime p, positive integers r,n, and a polynomial f with coefficients in 픽pr, let Wp,r,n(f) = fn 픽 pr ∖ fn+1 픽 pr. As n varies, the Wp,r,n(f) partition the set of strictly preperiodic points of the dynamical system induced by the action of f on 픽pr. In this paper, we compute statistics of strictly preperiodic points of dynamical systems induced by unicritical polynomials over finite fields by obtaining effective upper bounds for the proportion of 픽pr lying in a given Wp,r,n(f). Moreover, when we generalize our definition of Wp,r,n(f), we obtain both upper and lower bounds for the resulting averages. [ABSTRACT FROM AUTHOR]

Let R = 핂[x] be a univariate polynomial ring over an algebraically closed field 핂 of characteristic zero. Let A ϵ Mm,m(R) be an m x m matrix over R with non-zero determinate det(A) ϵ R. In this paper, utilizing linear-algebraic techniques, we investigate the relationship between a basis for the syzygy module of f1, . . ., fm and a basis for the syzygy module of g1, . . ., gm, where [g1, . . ., gm] = [f1, . . ., fm]A. [ABSTRACT FROM AUTHOR]

The fundamental aim of this research is to analyze the configuration of 픽4R submodules, skew cyclic codes over 픽4R and establish their connection with DNA codes, where 픽4 is a field of order 4 and R = 픽4 + u픽4 + v픽4 + w픽4 with u2 = u, v2 = v, w2 = w, uv = vu = 0, vw = wv = 0, wu = uw = 0 is a finite ring. This is achieved by examining particular subclasses like reversible codes. Ultimately, this study aims to utilize Gray maps to derive codes that possess the characteristics of DNA structures. At the end of this paper, we have provided the necessary and sufficient condition for skew cyclic codes to be reversible complement. [ABSTRACT FROM AUTHOR]

In this paper, we study linear codes invariant under a cyclic endomorphism T , called T -cyclic codes. Since every cyclic endomorphism can be represented by a cyclic matrix with respect to a given basis, all of these matrices are similar. For simplicity we restrict ourselves to linear codes invariant under the right multiplication by a cyclic matrix M , that we call M -cyclic codes, and when M is the companion matrix C f of a given nonzero polynomial f (x) we call them f -cyclic codes. The similarity relation between matrices helps us find connections between M -cyclic codes and f -cyclic codes, where f (x) is the minimal polynomial of M. The class of M -cyclic codes contains cyclic codes and their various generalizations such as constacyclic codes, right and left polycyclic codes, monomial codes, and others. As common in the study of cyclic codes and their generalizations, we make use of the one-to-one correspondence between M -cyclic codes and ideals of the polynomial ring R f : = q [ x ] / 〈 f (x) 〉 , where f (x) is the minimal polynomial of M. This correspondence leads to some basic characterizations of these codes such as generator and parity check polynomials among others. Next, we study the duality of these codes, where we show that the b -dual of an M -cyclic code is an M ∗ -cyclic code, where M ∗ is the adjoint matrix of M with respect to b , and we explore some important results on the duality of these codes. Finally, we give examples as applications of some of the results and we construct some optimal codes. [ABSTRACT FROM AUTHOR]

We resolve an open question posed by Nagpal, Sam and Snowden [Selecta. Math. (N.S.) 22 (2016), pp. 913–937] in 2015 concerning a Gröbner theoretic approach to the noetherianity of the twisted commutative algebra Sym(Sym^2(\mathbf {C}^\infty)). We provide a negative answer to their question by producing an explicit antichain. In doing so, we establish a connection to well-studied posets of graphs under the subgraph and induced subgraph relation. We then analyze this connection to suggest future paths of investigation. [ABSTRACT FROM AUTHOR]

We construct examples of bounded below, noncontractible, acyclic complexes of finitely generated projective modules over some rings S, as well as bounded above, noncontractible, acyclic complexes of injective modules. The rings S are certain rings of infinite matrices with entries in the rings of commutative polynomials or formal power series in infinitely many variables. In the world of comodules or contramodules over coalgebras over fields, similar examples exist over the cocommutative symmetric coalgebra of an infinite-dimensional vector space. A simpler, universal example of a bounded below, noncontractible, acyclic complex of free modules with one generator, communicated to the author by Canonaco, is included at the end of the paper. [ABSTRACT FROM AUTHOR]

Let S = K [ x 1 , ... , x n ] be a polynomial ring in n variables with coefficients over a field K. A t -spread lexsegment ideal I of S is a monomial ideal generated by a t -spread lexsegment set. We determine all t -spread lexsegment ideals with linear resolution by means of Betti splittings. As applications we provide formulas for the Betti numbers of such a class of ideals and furthermore we characterize all incompletely t -spread lexsegment ideals with linear quotients. [ABSTRACT FROM AUTHOR]

We demonstrate that the phenomenon of popular differences (aka the phenomenon of large intersections) holds for natural families of polynomial patterns in rings of integers of number fields. If K is a number field with ring of integers $\mathcal{O}_K$ and $E \subseteq \mathcal{O}_K$ has positive upper Banach density $d^*(E) = \delta > 0$ , we show, inter alia : (1) if $p(x) \in K[x]$ is an intersective polynomial (i.e., p has a root modulo m for every $m \in \mathcal{O}_K$) with $p(\mathcal{O}_K) \subseteq \mathcal{O}_K$ and $r, s \in \mathcal{O}_K$ are distinct and nonzero, then for any $\varepsilon > 0$ , there is a syndetic set $S \subseteq \mathcal{O}_K$ such that for any $n \in S$ , \begin{align*} d^* \left(\left\{ x \in \mathcal{O}_K \;:\; \{x, x + rp(n), x + sp(n)\} \subseteq E \right\} \right) > \delta^3 - \varepsilon. \end{align*} Moreover, if ${s}/{r} \in \mathbb{Q}$ , then there are syndetically many $n \in \mathcal{O}_K$ such that \begin{align*} d^* \left(\left\{ x \in \mathcal{O}_K \;:\; \{x, x + rp(n), x + sp(n), x + (r+s)p(n)\} \subseteq E \right\} \right) > \delta^4 - \varepsilon; \end{align*} (2) if $\{p_1, \dots, p_k\} \subseteq K[x]$ is a jointly intersective family (i.e., $p_1, \dots, p_k$ have a common root modulo m for every $m \in \mathcal{O}_K$) of linearly independent polynomials with $p_i(\mathcal{O}_K) \subseteq \mathcal{O}_K$ , then there are syndetically many $n \in \mathcal{O}_K$ such that \begin{align*} d^* \left(\left\{ x \in \mathcal{O}_K \;:\; \{x, x + p_1(n), \dots, x + p_k(n)\} \subseteq E \right\} \right) > \delta^{k+1} - \varepsilon. \end{align*} These two results generalise and extend previous work of Frantzikinakis and Kra [ 21 ] and Franztikinakis [ 19 ] on polynomial configurations in $\mathbb{Z}$ and build upon recent work of the authors and Best [ 2 ] on linear patterns in general abelian groups. The above combinatorial results follow from multiple recurrence results in ergodic theory via a version of Furstenberg's correspondence principle. The ergodic-theoretic recurrence theorems require a sharpening of existing tools for handling polynomial multiple ergodic averages. A key advancement made in this paper is a new result on the equidistribution of polynomial orbits in nilmanifolds, which can be seen as a far-reaching generalisation of Weyl's equidistribution theorem for polynomials of several variables: (3) let $d, k, l \in \mathbb{N}$. Let $(X, \mathcal{B}, \mu, T_1, \dots, T_l)$ be an ergodic, connected $\mathbb{Z}^l$ -nilsystem. Let $\{p_{i,j} \;:\; 1 \le i \le k, 1 \le j \le l\} \subseteq \mathbb{Q}[x_1, \dots, x_d]$ be a family of polynomials such that $p_{i,j}\left(\mathbb{Z}^d \right) \subseteq \mathbb{Z}$ and $\{1\} \cup \{p_{i,j}\}$ is linearly independent over $\mathbb{Q}$. Then the $\mathbb{Z}^d$ -sequence $\left(\prod_{j=1}^l{T_j^{p_{1,j}(n)}}x, \dots, \prod_{j=1}^l{T_j^{p_{k,j}(n)}}x \right)_{n \in \mathbb{Z}^d}$ is well-distributed in $X^k$ for every x in a co-meager set of full measure. [ABSTRACT FROM AUTHOR]

A foundational principle in the study of modules over standard graded polynomial rings is that geometric positivity conditions imply vanishing of Betti numbers. The main goal of this paper is to determine the extent to which this principle extends to the nonstandard Z$\mathbb {Z}$‐graded case. In this setting, the classical arguments break down, and the results become much more nuanced. We introduce a new notion of Castelnuovo–Mumford regularity and employ exterior algebra techniques to control the shapes of nonstandard Z$\mathbb {Z}$‐graded minimal free resolutions. Our main result reveals a unique feature in the nonstandard Z$\mathbb {Z}$‐graded case: the possible degrees of the syzygies of a graded module in this setting are controlled not only by its regularity, but also by its depth. As an application of our main result, we show that given a simplicial projective toric variety and a module M$M$ over its coordinate ring, the multigraded Betti numbers of M$M$ are contained in a particular polytope when M$M$ satisfies an appropriate positivity condition. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in linear codes invariant under the (σ , δ) -action of a given matrix M , that we call (M , σ , δ) -skew codes. Various subclasses of these codes, depending on the (σ , δ) -cyclical property of M , are derived. In particular, we distinguish between three fundamental types: (M , σ , δ) -cyclic, (M , σ , δ) -quasi-cyclic and (M , σ , δ) -multi-cyclic. These different subclasses allow us to obtain some of the well known types of linear codes as special cases depending on the choice of M. We observe that the (σ , δ) -similarity in M n (F q ) gives rise to various relations between these codes. As common in the study of cyclic codes and their generalizations, we use the skew polynomial ring F q [ x , σ , δ ] to study (M , σ , δ) -skew codes. As an application of the theory developed here, we construct some optimal linear codes and MDS codes over different alphabets with additional desirable properties such as self orthogonality and linear complementary duality (LCD). [ABSTRACT FROM AUTHOR]

Let G = (V , E) be a simple graph with vertex set as V and edge set as E. Within the framework of graph-based binary coding, a graph G is a threshold graph if and only if it can be generated by a binary code of the type 0 s 1 1 t 1 ... 0 s k 1 t k , where s i and t i are positive integers with 1 ≤ i ≤ k. This type of binary code is called a binary generating code of G. In this paper, we address a problem originally posed by Harary and Schwenk in [10] titled "Which graphs have integral spectra?" in which the main focus of our investigations lies in the Seidel Laplacian spectrum of G generated by 0 s 1 1 t 1 ... 0 s k 1 t k . We establish that all eigenvalues of G are integers and multiplicities of eigenvalues are equal to powers of bits 0 and 1 (sizes of strings of 0's and 1's) involved in a binary generating code of G. We exhibit a necessary and sufficient condition in terms of eigenvalues of the matrix associated with a graph-based G binary code to be Seidel Laplacian integral. Furthermore, for any prime p , we explore the Seidel Laplacian spectrum of zero-divisor graphs associated with the ring Z p m (where m is a positive integer) and the polynomial ring Z p [ x ] / (x p). [ABSTRACT FROM AUTHOR]

Let R : = F [ x ; σ , δ ] be a multivariate skew polynomial ring over a division ring F. In this paper, we introduce the notion of right and left (σ , δ) -partial derivatives of polynomials in R and we prove some of their main properties. As an application of these results, we solve in R a Hermite-type multivariate skew polynomial interpolation problem. The main technical tools and results used here are of constructive type, showing methods and algorithms to construct a polynomial in R which satisfies the above Hermite-type interpolation problem and its relative Lagrange-type version. [ABSTRACT FROM AUTHOR]

In this paper, we establish an explicit higher reciprocity law for the polynomial ring over a nonprincipal ultraproduct of finite fields. Such an ultraproduct can be taken over the same finite field, which allows to recover the classical higher reciprocity law for the polynomial ring F q [ t ] over a finite field F q that is due to Dedekind, Kühne, Artin, and Schmidt. On the other hand, when the ultraproduct is taken over finite fields of unbounded cardinalities, we obtain an explicit higher reciprocity law for the polynomial ring over an infinite field in both characteristics 0 and p > 0 for some prime p. We then use the higher reciprocity law to prove an analogue of the Grunwald–Wang theorem for such a polynomial ring in both characteristics 0 and p > 0 for some prime p. [ABSTRACT FROM AUTHOR]

We present an algorithm to determine the realization of matrices in SL2 over the multivariate Laurant polynomial ring R[x± 1, ..., x± k ]. For this we have to generalize Park's algorithm. Thus the purpose is to express a matrix in SL2(R[x± 1, ..., x± k ]) as a product of elementary matrices. Furthermore, we extend this algorithm to some specific matrices in SL3(R[x± 1, ..., x± k ]). To illustrate our results, some examples are studied using special software which implements the proposed algorithms. [ABSTRACT FROM AUTHOR]

In lattice-based cryptography, Ring Learning with Errors (RLWE) is a computationally hard cryptographic problem, comprising three basic mechanisms i.e., key generation, encryption, and decryption. Binary Ring Learning with Error (BRLWE), a new variant of RLWE has been proposed recently to reduce the key size and computational complexity compared to previous RLWE-based schemes. Based on this BRLWE scheme, efficient hardware architectures have been obtained in recent works for lightweight applications. The key operation involved in this scheme is A B + C , where A and C are integer polynomials and B is a binary polynomial. This paper proposes an efficient hardware architecture for BRLWE-based scheme targeted for lightweight applications. The architecture computes the arithmetic operation A B + C , which includes polynomial multiplication and addition over the polynomial ring Z q / (x n + 1). The proposed architecture is applied in two conditions, fixed and variable values of q. Experimental results show the architecture proposed has 50% less Area-Delay Product (ADP) and 20% less Power-Delay Product (PDP) compared to the recently reported work for n = 256. [ABSTRACT FROM AUTHOR]

Let F q [ t ] be the polynomial ring over the finite field F q of q elements. For 1 ≤ j ≤ l , let a j , a j ′ ∈ F q [ t ] with a j ≠ 0. Let H (x) denote the product of l linear functions a j x + a j ′. Let X (H) be the set of all monic polynomials n in F q [ t ] such that H (n) is the product of l monic irreducible polynomials in F q [ t ]. The Hardy-Littlewood l -tuple conjecture provides an asymptotic formula for the cardinality of the set A N : = { n ∈ X (H) : deg ⁡ n = N } when N tends to ∞. In this paper an F q [ t ] -version of the enveloping sieve theory is established by following the constructing method of Green-Tao. It is applied to give an upper bound for the cardinality of ⋃ j < N A j. It is also applied to obtain Lebesgue space estimates for the exponential sum over X (H). These results are the best possible if the Hardy-Littlewood l -tuple conjecture is true. [ABSTRACT FROM AUTHOR]

For the secure deployment of network platforms tailored for IoT devices, the encryption of data transmission is equally as crucial as the process of authentication. In this context, we introduce the Secure and Scalable IoT network (SSI) network platform, designed to accommodate a diverse range of IoT devices. It provides scalability and implements effective many-to-many and end-to-end encryption across extensive regions. With the emergence of quantum computing, secure public key exchange mechanisms have become important. Among the various post-quantum cryptography (PQC) algorithms assessed, Nth Degree Truncated Polynomial Ring Units (NTRUs) have emerged as an optimally suited PQC algorithm for IoT devices constrained by limited computational capabilities. We have integrated NTRUs with SSI as a lightweight PQC solution. Moreover, SSI-PQM (Post-Quantum MACsec) enhances the SSI's initial authentication structure to minimize PQC-TLS session attempts and protect the SSI's important configuration information. When applying TLS with PQC for secret key exchange purposes, it was verified that this approach ensures stable performance in IoT environments. Upon the implementation of our proposed SSI-PQM on Raspberry Pi 3B+ based IoT devices, SSI-PQM exhibited acceptable performance at security levels from 80 to 128 and achieved a minimum speed improvement of 161% over RSA at security levels above 160. It can be concluded that SSI-PQM stands out as an effective Zero Trust-based IoT network platform, demonstrating its viability and efficiency in safeguarding data transmission against potential quantum computing threats. [ABSTRACT FROM AUTHOR]

Let D be a Dedekind domain with finite residue fields. We provide topological insights into certain classes of ideals of Int(D) lying over a given maximal ideal \mathfrak m of D. We completely determine invertible/divisorial ideals in terms of topological properties of subsets of the \mathfrak m-adic completion of D. Moreover, these results are naturally extended to overrings of Int(D). As an application we provide explicit constructions of divisorial ideals of Int(D) which are not finitely generated. [ABSTRACT FROM AUTHOR]

Say that (x, y, z) is a positive primitive integral Pythagorean triple if x, y, z are positive integers without common factors satisfying x^2 + y^2 = z^2. An old theorem of Berggren gives three integral invertible linear transformations whose semi-group actions on (3, 4, 5) and (4, 3, 5) generate all positive primitive Pythagorean triples in a unique manner. We establish an analogue of Berggren's theorem in the context of a one-variable polynomial ring over a field of characteristic \neq 2. As its corollaries, we obtain some structure theorems regarding the orthogonal group with respect to the Pythagorean quadratic form over the polynomial ring. [ABSTRACT FROM AUTHOR]