Polynomial Ring Research Articles

Error-correcting codes for fermionic quantum simulation

Utilizing the framework of \mathbb{Z}_2ℤ2 lattice gauge theories in the context of Pauli stabilizer codes, we present methodologies for simulating fermions via qubit systems on a two-dimensional square lattice. We investigate the symplectic automorphisms of the Pauli module over the Laurent polynomial ring. This enables us to systematically increase the code distances of stabilizer codes while fixing the rate between encoded logical fermions and physical qubits. We identify a family of stabilizer codes suitable for fermion simulation, achieving code distances of d=2,3,4,5,6,7, allowing correction of any \lfloor \frac{d-1}{2} \rfloor⌊d−12⌋-qubit error. In contrast to the traditional code concatenation approach, our method can increase the code distances without decreasing the (fermionic) code rate. In particular, we explicitly show all stabilizers and logical operators for codes with code distances of d=3,4,5. We provide syndromes for all Pauli errors and invent a syndrome-matching algorithm to compute code distances numerically.

Factorization properties of quotients of polynomial and power series rings by monomial ideals

Suppose that K is a field, S = K [ x 1 , … , x n ] or S = K [ [ x 1 , … , x n ] ] and I is a monomial ideal of S. We study factorization properties of the ring R = S / I . In particular, we present conditions equivalent to R being présimplifiable, a bounded factorization ring, a finite factorization ring or a unique factorization ring.

Performance and Efficiency Exploration of Hardware Polynomial Multipliers for Post-Quantum Lattice-Based Cryptosystems

The significant effort in the research and design of large-scale quantum computers has spurred a transition to post-quantum cryptographic primitives worldwide. The post-quantum cryptographic primitive standardization effort led by the US NIST has recently selected the asymmetric encryption primitive Kyber as its candidate for standardization and indicated NTRU, as a valid alternative if intellectual property issues are not solved. Finally, a more conservative alternative to NTRU, NTRUPrime was also considered as an alternate candidate, due to its design choices that remove the possibility for a large set of attacks preemptively. All the aforementioned asymmetric primitives provide good performances, and are prime choices to provide IoT devices with post-quantum confidentiality services. In this work, we present a comprehensive exploration of hardware designs for the computation of polynomial multiplications, the workhorse operation in all the aforementioned cryptosystems, with a thorough analysis of performance, compactness and efficiency. The presented designs cope with the differences in the arithmetics of polynomial rings employed by distinct cryptosystems, benefiting from configurations and optimizations that are applicable at synthesis time and/or run time. In this context, we target a use case scenario where long-term key pairs are used, such as the ones for VPNs (e.g., over IPSec), secure shell protocols and instant messaging applications. Our high-performance design variants exhibit figures of latency comparable to the ones needed for the execution of the symmetric cryptographic primitives also included in the Post-Quantum schemes. Notably, the performance figures of the designs proposed for NTRU and NTRU Prime surpass the ones described in the related literature.

On the primitive subspace of Lando framed graph bialgebra

Lando framed graph bialgebra is generated by framed graphs modulo 4-term relations. We provide an explicit set of generators of its primitive subspace and a description of the set of relations between the generators. We also define an operation of leaf addition that endows the primitive subspace of Lando algebra with a structure of a module over the ring of polynomials in one variable and construct a 4-invariant that satisfies a simple identity with respect to the vertex-multiplication.

Computing the binomial part of a polynomial ideal

Given an ideal I in a polynomial ring K[x1,…,xn] over a field K, we present a complete algorithm to compute the binomial part of I, i.e., the subideal Bin(I) of I generated by all monomials and binomials in I. This is achieved step-by-step. First we collect and extend several algorithms for computing exponent lattices in different kinds of fields. Then we generalize them to compute exponent lattices of units in 0-dimensional K-algebras, where we have to generalize the computation of the separable part of an algebra to non-perfect fields in characteristic p. Next we examine the computation of unit lattices in finitely generated K-algebras, as well as their associated characters and lattice ideals. This allows us to calculate Bin(I) when I is saturated with respect to the indeterminates by reducing the task to the 0-dimensional case. Finally, we treat the computation of Bin(I) for general ideals by computing their cellular decomposition and dealing with finitely many special ideals called (s,t)-binomial parts. All algorithms have been implemented in SageMath.

Noncommutative Coding Theory and Algebraic Sets for Skew PBW Extensions

The classical commutative coding theory has been recently extended to noncommutative rings of polynomial type. There are many interesting works in coding theory over single Ore extensions. In this review article we present the most relevant algebraic tools and properties of single Ore extensions used in noncommutative coding theory. The last section represents the novelty of the paper. We will discuss the algebraic sets arising in noncommutative coding theory but for skew PBW extensions. These extensions conform a general class of noncommutative rings of polynomial type and cover several algebras arising in physics and noncommutative algebraic geometry, in particular, they cover the Ore extensions of endomorphism injective type and the polynomials rings over fields.

The Theory of the Entire Algebraic Functions

Abstract Let $A$ be the integral closure of the ring of polynomials ${{\mathbb {C}}}[t]$, within the field of algebraic functions in one variable. We show that $A$ interprets the ring of integers. This contrasts with the analogue for finite fields, proved to have a decidable theory in [12] and [4].

Deformation cohomology for cyclic groups acting on polynomial rings

We examine the Hochschild cohomology governing graded deformations for finite cyclic groups acting on polynomial rings. We classify the infinitesimal graded deformations of the skew group algebra S⋊G for a cyclic group G acting on a polynomial ring S. This gives all graded deformations of the first order. We are particularly interested in the case when the characteristic of the underlying field divides the order of the acting group, which complicates the determination of cohomology.

Normality of Rees algebras of generalized mixed product ideals

Let $K$ be a field and $K[x_1,x_{2}]$ the polynomial ring in two
 variables over $K$ with each $x_i$ of degree $1$. Let $L$ be the
 generalized mixed product ideal induced by a monomial ideal
 $I\subset K[x_1,x_2]$, 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 $\mathcal{R}(L)$, the
 Rees algebra of $L$. Furthermore, we give a geometric description of
 the integral closure of $\mathcal{R}(L)$.

Computational methods for t-spread monomial ideals

Let $K$ be a field and $S=K[x_1,\ldots,x_n]$ 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 \emph{Macaulay2} package, \texttt{TSpreadIdeals}, to ease the computation of well-known results about algebraic invariants for $t$-spread ideals.

A fast NTRU software implementation based on 5-way TMVP

The fast and efficient operation of post-quantum cryptographic algorithms has become an important research topic, especially with the recent developments and the process of the PQC Standardization Project by NIST. Various algorithms based on lattices are chosen to be finalists in this project. Number Theoretic Transform (NTT), Toom–Cook, Karatsuba, and Toeplitz matrix–vector product (TMVP) are considered in order to efficiently perform multiplications in quotient polynomial rings required by the cryptographic schemes based on lattices. In this paper, we propose a 5-way split TMVP-based algorithm for multiplication in polynomial quotient rings and determine its computational complexity. The new algorithm is derived from a 5-term polynomial multiplication algorithm using 13 multiplications with coefficients of only 1 or -1 that provide efficiency. We also implement it for all parameter sets of NTRU. The results show that we have up to 34%, 35%, and 157% speed-up against Toom4-Karatsuba implementation in key generation, encapsulation, and decapsulation, respectively.

Необходимое и достаточное условия существования рациональных решений однородных разностных уравнений с постоянными коэффициентами

A necessary and a sufficient condition for solvability of homogeneous difference equations with constant coefficients in the class of rational functions are obtained. The necessary condition is a restriction on the Newton polyhedron of the characteristic polynomial. In the two-dimensional case, this condition is the existence of parallel sides on the polygon. The sufficient condition is the equality to zero of certain sums of the coefficients of the equation. If the sufficient condition is satisfied, the solution is the class of rational functions whose denominators form a subring in the ring of polynomials. This subring can be associated with an edge of the Newton polyhedron of the characteristic polynomial of the equation.

S‐J‐Ideals: A Study in Commutative and Noncommutative Rings

In this paper, we introduce the concept of S‐‐ideals in both commutative and noncommutative rings. For a commutative ring R and a multiplicatively closed subset S, we show that many properties of ‐ideals apply to S‐‐ideals and examine their characteristics in various ring constructions, such as homomorphic image rings, quotient rings, cartesian product rings, polynomial rings, power series rings, idealization rings, and amalgamation rings. In noncommutative rings, where S is an m‐system, we define right S‐‐ideals. We demonstrate the equivalence of S‐‐ideals and right S‐‐ideals in commutative rings with identity and provide examples to illustrate the connections between right S‐prime ideals and ‐ideals.

Towards the Computation of Stabilizing Controllers of Multidimensional Systems

In this paper, we further study the effective computation of stabilizing controllers of multidimensional systems. Within the algebraic analysis approach, the stabilization problem can be characterized by the fact that a certain finitely presented A-module M, naturally associated with the multidimensional system, is projective, where A denotes the ring of multivariate rational functions without poles in the closed complex unit polydisc Dn. This condition can be reduced to the existence of an element s of a polynomial ideal I which has no zero in Dn. According to the Polydisc Nullstellensatz, the latter condition is equivalent to the fact that no complex zero of the elements of I belongs to Dn. If this condition is satisfied, using cyclic resultants and linear programming, we then propose a method to compute such a polynomial s. Finally, using computer algebra methods for effectively handling basic operations on R[s-1]-modules, where R is a polynomial ring, we show how to compute stabilizing controllers.

Increasing subsequences, matrix loci and Viennot shadows

Abstract Let ${\mathbf {x}}_{n \times n}$ be an $n \times n$ matrix of variables, and let ${\mathbb {F}}[{\mathbf {x}}_{n \times n}]$ be the polynomial ring in these variables over a field ${\mathbb {F}}$ . We study the ideal $I_n \subseteq {\mathbb {F}}[{\mathbf {x}}_{n \times n}]$ generated by all row and column variable sums and all products of two variables drawn from the same row or column. We show that the quotient ${\mathbb {F}}[{\mathbf {x}}_{n \times n}]/I_n$ admits a standard monomial basis determined by Viennot’s shadow line avatar of the Schensted correspondence. As a corollary, the Hilbert series of ${\mathbb {F}}[{\mathbf {x}}_{n \times n}]/I_n$ is the generating function of permutations in ${\mathfrak {S}}_n$ by the length of their longest increasing subsequence. Along the way, we describe a ‘shadow junta’ basis of the vector space of k-local permutation statistics. We also calculate the structure of ${\mathbb {F}}[{\mathbf {x}}_{n \times n}]/I_n$ as a graded ${\mathfrak {S}}_n \times {\mathfrak {S}}_n$ -module.

Конструирование гладких выпуклых продолжений булевых функций

Systems of Boolean equations are widely used in mathematics, computer science, and applied sciences. In this regard, on the one hand, new research methods and algorithms are being developed for such systems, and on the other hand, existing methods and algorithms for solving such systems are being improved. One of these methods is that, firstly, the system of Boolean equations given over the ring of Boolean polynomials is transformed into a system of equations over the field of real numbers, and secondly, the transformed system is reduced either to the problem of numerical minimization of the corresponding objective function, to a MILP or QUBO problem, to a system of polynomial equations solved on the set of integers, or to an equivalent system of polynomial equations solved by symbolic methods. There are many ways to transform a system of Boolean equations into a continuous minimization problem, since the fundamental difference between such methods and “brute force” local search algorithms is that at each iteration of the algorithm, the shift along the antigradient is performed on all variables simultaneously. But one of the main problems that arise when applying these methods is that the objective function to be minimized in the desired area can have many local minima, which greatly complicates their practical use. In this paper, a non-negative convex and continuously differentiable extension of any Boolean function is constructed, which is applied to solving an arbitrary system of Boolean equations. It is argued that the problem of solving an arbitrary system of Boolean equations can be constructively reduced to the problem of minimizing a function, any local minimum of which in the desired domain is a global minimum.

The signature of a monomial ideal

<p>The irreducible decomposition of a monomial ideal has played an important role in combinatorial commutative algebra, with applications beyond pure mathematics, such as biology. Given a monomial ideal $ I $ of a polynomial ring $ S = k[{\bf x}] $ over a field $ k $ and variables $ {\bf x} = \{x_1, \ldots, x_n\} $, its incidence matrix, is the matrix whose rows are indexed by the variables $ {\bf x} $ and whose columns are indexed by its minimal generators. The main contribution of this paper is the introduction of a novel invariant of a monomial ideal $ I $, termed its signature, which could be thought of as a type of canonical form of its incidence matrix, and the proof that two monomial ideals with the same signature have essentially the same irreducible decomposition.</p>

Finite intersection of valuation overrings of polynomial rings in at most three variables

The group of divisibility of an integral domainis the multiplicative group of nonzero principal fractional ideals ofthe domain and is a partially ordered group under reverse inclusion. We study the group of divisibility of a finite intersection of valuation overrings of polynomial rings in at most three variables and we classify all semilocal lattice-ordered groups which are realizable over k[x1,x2,...,xn] for n≤3.

A note on the degree bounds of the invariant ring

<abstract><p>Let $ G = C_p\times H $ be a finite group, where $ C_p $ is a cyclic group of prime order $ p $ and $ H $ is a $ p^{\prime} $-group. Let $ \mathbb{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 $ \mathbb{F}[V] $ with actions of $ G $. In this paper, by the periodicity property, we showed that $ \mathbb{F}[V]^G $ is generated by $ m $ norm polynomials together with homogeneous invariants of degree at most $ m|G|-{\rm dim}(V) $ and transfer invariants, which yields the well-known degree bound $ {\rm dim} $$ (V)\cdot(|G|-1) $. More precisely, we found that this bound gets less sharp as the dimensions of simple $ H $-modules increase.</p></abstract>

Equivariant resolutions over Veronese rings

AbstractWorking in a polynomial ring , where is an arbitrary commutative ring with 1, we consider the th Veronese subalgebras , as well as natural ‐submodules inside . We develop and use characteristic‐free theory of Schur functors associated to ribbon skew diagrams as a tool to construct simple ‐equivariant minimal free ‐resolutions for the quotient ring and for these modules . These also lead to elegant descriptions of for all and for any pair of these modules .