Search

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 article, we present the generalized ρ dependent polynomials for the calculations of eccentricity, distance, total distance, and degree-based topological indices of the identity graph of Zρ. This is a thorough work in which we present many topological indices and coindices as ρ dependent polynomials. The polynomials presented here can play a key role in the further development of the theory of topological indies for commutative rings. This paper presents a brand new approach to generalizing the topological indices because instead of the traditional way. A set-theoretic method is introduced here that can be very helpful and game-changing in the field of algebraic graph theory first of all sets of vertices for the identity graph of the commutative ring Zρ are partitioned into various sets, which makes it easier to generalize the degrees, distances, and eccentricities of this graph. This paper presents various results that make it easier to generalize the topological indices of the identity graph of Zρ. [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]

Copyright of Journal of Cryptologic Research (2095-7025) is the property of Editorial Board of Journal of Cryptologic Research and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

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]

Let Pn := H*((RP∞)n) =≈ Z2[x1, x2, ..., xn] be the graded polynomial algebra over K, where K denotes the prime field of two elements. We investigate the Peterson hit problem for the polynomial algebra Pn, viewed as a graded left module over the mod-2 Steenrod algebra, A. For n > 4, this problem is still unsolved, even in the case of n = 5 with the help of computers. In this paper, we study the hit problem for the case n = 6 in degree dk = 6(2k - 1) + 9.2k, with k an arbitrary non-negative integer. By considering K as a trivial A-module, then the hit problem is equivalent to the problem of finding a basis of K-graded vector space KAPn. The main goal of the current paper is to explicitly determine an admissible monomial basis of the K-graded vector space KAP6 in some degrees. At the same time, the behavior of the sixth Singer algebraic transfer in degree dk = 6(2k - 1) + 9.2k is also discussed at the end of this article. Here, the Singer algebraic transfer is a homomorphism from the homology of the mod-2 Steenrod algebra, Torn,n+dA(K,K), to the subspace of K A Pn consisting of all the GLn-invariant classes of degree d. [ABSTRACT FROM AUTHOR]

Let G be a finite simple graph and I (G) denote the corresponding edge ideal in a polynomial ring over a field . In this paper, we obtain upper bounds for the Castelnuovo–Mumford regularity of symbolic powers of certain classes of edge ideals. We also prove that for several classes of graphs, the regularity of symbolic powers of their edge ideals coincides with that of their ordinary powers. [ABSTRACT FROM AUTHOR]

Let V be a valuation ring such that dim(V) = 0 and the annihilator of each element in V is finitely generated. In this paper it is proved that if I is a finitely generated ideal in the polynomial ring V [X], then there is a Gröbner basis for I. Also, an example of a zero-dimensional non-Noetherian valuation ring RM is presented, together with an example of finding a Gröbner basis for a certain ideal in a polynomial ring RM[X]. [ABSTRACT FROM AUTHOR]

Let 풜 = (Rn)n≥0 be an ascending chain of commutative rings with identity and let 풜[X] be the ring of polynomials with coefficient of degree i in Ri for each i ∈ ℕ. In the first part of this paper, we give a necessary and sufficient conditions for the ring 풜[X] to satisfy Noetherian spectrum (a ring R is said to have Noetherian spectrum if R satisfies the Ascending Chain Condition (ACC) on radical ideals). In particular, we extend classical results proved by Ohm and Pendleton in [J. Ohm and R. Pendleton, Rings with Noetherian spectrum, Duke Math. J. 35 (1968) 631–639] and by Hizem in [S. Hizem, Chain conditions in rings of the form A + XB[X] and A + XI[X], in Commutative Algebra and its Applications (de Gruyter, 2009), pp. 259–274]. The second part of this paper is motivated by the interesting results proved on uniformly S-Noetherian rings by Kim et al. in [W. Qi, H. Kim, F. Wang, M. Chen and W. Zhao, Uniformly S-Noetherian rings (submitted)]. We introduce the concept of uniformly S-Noetherian spectrum and study its properties. Let R be a commutative ring and S a multiplicative subset of R. We say that R has uniformlyS-Noetherian spectrum if that there exists an s ∈ S such that for any ideal I of R, sI ⊆J for some finitely generated subideal J of I. We give the Eakin–Nagata–Formanek theorem for the uniformly S-Noetherian spectrum condition. We also study the uniformly S-Noetherian uniformly properties on several ring constructions (Nagata’s idealization and amalgamated algebras). [ABSTRACT FROM AUTHOR]

The algebraic structure of skew cyclic codes over M2(F2), using the F4-cyclic algebra, is studied in this work. We determine that a skew cyclic code with a polynomial of minimum degree d(x) is a free code generated by d(x). According to our findings, skew cyclic codes of odd and even lengths are cyclic and 2-quasi-cyclic over M2(F2), respectively. We provide the self-dual skew condition of Hermitian dual codes of skew cyclic codes. The generator polynomials of Euclidean dual codes are obtained. Furthermore, a spanning set of a double skew cyclic code over M2l(F2) is considered in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in monomial codes with associated vector a = (a0, a1,..., an-1), introduced in [4], and more generally in linear codes invariant under a monomial matrix M = diag(a0,a1,...,an-1)Pσ where σ is a permutation and Pσ its associated permutation matrix. We discuss some connections between monomial codes and codes invariant under an arbitrary monomial matrix M. Next, we identify monomial codes with associated vector a = (a0,a2,...,an-1) by the ideals of the polynomial ring ..., via a special isomorphism ... which preserves the Hamming weight and differs from the classical isomorphism used in the case of cyclic codes and their generalizations. This correspondence leads to some basic characterizations of monomial codes such as generator polynomials, parity check polynomials, and others. Next, we focus on the structure of l--quasi-monomial (l--QM) codes of length n = ml, where on the one hand, we characterize them by the Rq,m --submodules of Rq,ml . On the other hand, l--QM codes are seen as additive monomial codes over the extension 픽ql/Fq. So, as in the case of quasi-cyclic codes [8], we characterize those codes that have 픽ql--linear images with respect to a basis of the extension 픽ql/Fq, based on the CRT decomposition. Finally, we show that l--QM codes and additive monomial codes are asymptotically good. [ABSTRACT FROM AUTHOR]

In this paper, we consider rings of multisets consisting of elements of a Banach algebra. We investigate the algebraic and topological structures of such rings and the properties of their homomorphisms. The rings of multisets arise as natural domains of supersymmetric functions. We introduce a complete metrizable topology on a given ring of multisets and extend some known results about structures of the rings to the general case. In addition, we consider supersymmetric polynomials and other supersymmetric functions related to these rings. This paper contains a number of examples and some discussions. [ABSTRACT FROM AUTHOR]

DBTRU was proposed by Thang and Binh in 2015. As a variant of NTRU, the integer polynomial ring is replaced by two binary truncated polynomial rings G F (2) [ x ] / (x n + 1) . DBTRU has some advantages over NTRU in terms of security and performance. In this paper, we propose a polynomial-time linear algebra attack against the DBTRU cryptosystem, which can break DBTRU for all recommended parameter choices. The paper shows that the plaintext can be achieved in less than 1 s via the linear algebra attack on a single PC. [ABSTRACT FROM AUTHOR]

In this paper, we extend work from [M. Vancliff and P. P. Veerapen, Generalizing the notion of rank to noncommutative quadratic forms, in Noncommutative Birational Geometry, Representations and Combinatorics, eds. A. Berenstein and V. Retakh, Contemporary Mathematics, Vol. 592 (2013), pp. 241–250], where a notion of rank, called μ -rank, was proposed for noncommutative quadratic forms on two and three generators. In particular, we provide a definition of μ -rank one and two for noncommutative quadratic forms on four generators. We apply this definition to determine the number of point modules over certain quadratic AS-regular algebras of global dimension four. [ABSTRACT FROM AUTHOR]

In this paper, we continue the discussion about relations between exponential polynomials and generalized moment functions on a commutative hypergroup. We are interested in the following problem: is it true that every finite-dimensional variety is spanned by moment functions? Let m be an exponential on X. In our former paper, we have proved that if the linear space of all m-sine functions in the variety of an m-exponential monomial is (at most) one-dimensional, then this variety is spanned by moment functions generated by m. In this paper, we show that this may happen also in cases where the m-sine functions span a more than one-dimensional subspace in the variety. We recall the notion of a polynomial hypergroup in d variables, describe exponentials on it and give the characterization of the so-called m-sine functions. Next we show that the Fourier algebra of a polynomial hypergroup in d variables is the polynomial ring in d variables. Finally, using the Ehrenpreis–Palamodov Theorem, we show that every exponential polynomial on the polynomial hypergroup in d variables is a linear combination of moment functions contained in its variety. [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]

In this paper, we show that there is no polynomial D(n)-quadruple in ℤ[X] for some polynomials n ∈ ℤ[X] that are not representable as a difference of squares of two polynomials in ℤ[X]. [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]