Multivariate Polynomial Rings Research Articles

On the realizability of the special linear group over the multivariate Laurent polynomial ring

On the realizability of the special linear group over the multivariate Laurent polynomial ring

Construction of quantum codes from multivariate polynomial rings

Construction of quantum codes from multivariate polynomial rings

The multivariate Serre conjecture ring

It is well-known that for any integral domain R, the Serre conjecture ring R〈X〉, i.e., the localization of the univariate polynomial ring R[X] at monic polynomials, is a Bézout domain of Krull dimension ≤1 if R is a Bézout domain of Krull dimension ≤1. Consequently, defining by induction R〈X1,…,Xn〉:=(R〈X1,…,Xn−1〉)〈Xn〉, the ring R〈X1,…,Xn〉 is a Bézout domain of Krull dimension ≤1 if so is R. The fact that R〈X1,…,Xn〉 is a Bézout domain when R is a valuation domain of Krull dimension ≤1 was the cornerstone of Brewer and Costa's theorem stating that if R is a one-dimensional arithmetical ring then finitely generated projective R[X1,…,Xn]-modules are extended. It is also the key of the proof of the Gröbner Ring Conjecture in the lexicographic order case, namely the fact that for any valuation domain R of Krull dimension ≤1, any n∈N>0, and any finitely generated ideal I of R[X1,…,Xn], the ideal LT(I) generated by the leading terms of the elements of I with respect to the lexicographic monomial order is finitely generated. Since the ring R〈X1,…,Xn〉 can also be defined directly as the localization of the multivariate polynomial ring R[X1,…,Xn] at polynomials whose leading coefficients according to the lexicographic monomial order with X1<X2<⋯<Xn is 1, we propose to generalize the fact that R〈X1,…,Xn〉 is a Bézout domain of Krull dimension ≤1 when R is a Bézout domain of Krull dimension ≤1 to any rational monomial order, bolstering the evidence for the Gröbner Ring Conjecture in the rational case. We give an example showing that this result is no more true in the irrational case.

Symmetry in multivariate ideal interpolation

An interpolation problem is defined by a set of linear forms on the (multivariate) polynomial ring and values to be achieved by an interpolant. For Lagrange interpolation the linear forms consist of evaluations at some nodes, while Hermite interpolation also considers the values of successive derivatives. Both are examples of ideal interpolation in that the kernels of the linear forms intersect into an ideal. For an ideal interpolation problem with symmetry, we address the simultaneous computation of a symmetry adapted basis of the least interpolation space and the symmetry adapted H-basis of the ideal. Beside its manifest presence in the output, symmetry is exploited computationally at all stages of the algorithm. For an ideal invariant, under a group action, defined by a Gröbner basis, the algorithm allows to obtain a symmetry adapted basis of the quotient and of the generators. We shall also note how it applies surprisingly but straightforwardly to compute fundamental invariants and equivariants of a reflection group.

Chinese remainder theorem of modules over multivariate polynomial rings

The Chinese remainder theorem plays an important role in the areas of number theory and algebra. In the multivariate polynomial rings, the Chinese remainder theorem can be used to solve systems of multivariate polynomial congruence equations. With the help of algebra theory, polynomials that satisfy the congruence equations can be found. In this paper, we investigate the Chinese remainder theorem of modules over multivariate polynomial rings. Using the theory and method of Gröbner bases for modules, we find the polynomial vector that satisfies the Chinese remainder theorem of modules. Furthermore, we obtain the solutions of systems of congruence equations for modules over polynomial rings.

A Combinatorial Model for the Decomposition of Multivariate Polynomial Rings as $S_n$-Modules

We consider the symmetric group $S_n$-module of the polynomial ring with $m$ sets of $n$ commuting variables and $m'$ sets of $n$ anti-commuting variables and show that the multiplicity of an irreducible indexed by the partition $\lambda$ (a partition of $n$) is the number of multiset tableaux of shape $\lambda$ satisfying certain column and row strict conditions. We also present a finite generating set for the ring of $S_n$ invariant polynomials of this ring.

On certain polynomial systems involving Stirling numbers of second kind

We solve a special type of linear systems with coefficients in multivariate polynomial rings. These systems arise in the computation of b-functions with respect to weights of certain hypergeometric ideals in the Weyl algebra.

The trailing terms ideal

In this paper, we address the following question: for a nonzero finitely generated ideal [Formula: see text] of a multivariate polynomial ring [Formula: see text] over a coherent ring [Formula: see text], fixing a monomial order [Formula: see text] on [Formula: see text], is the trailing terms ideal [Formula: see text] of [Formula: see text] (that is, the ideal generated by the trailing terms of the nonzero polynomials in [Formula: see text]) finitely generated? We show that while [Formula: see text] can be nonfinitely generated, it is always countably generated when the monomial order is Noetherian (graded monomial orders as instances).

A Signature-Based Algorithm for Computing Gr\xf6bner Bases over Principal Ideal Domains

Signature-based algorithms have become a standard approach for Gröbner basis computations for polynomial systems over fields, but how to extend these techniques to coefficients in general rings is not yet as well understood. In this paper, we present a proof-of-concept signature-based algorithm for computing Gröbner bases over commutative integral domains. It is adapted from a general version of Möller’s algorithm (J Symb Comput 6(2–3), 345–359, 1988) which considers reductions by multiple polynomials at each step. This algorithm performs reductions with non-decreasing signatures, and in particular, signature drops do not occur. When the coefficients are from a principal ideal domain (e.g. the ring of integers or the ring of univariate polynomials over a field), we prove correctness and termination of the algorithm, and we show how to use signature properties to implement classic signature-based criteria to eliminate some redundant reductions. In particular, if the input is a regular sequence, the algorithm operates without any reduction to 0. We have written a toy implementation of the algorithm in Magma. Early experimental results suggest that the algorithm might even be correct and terminate in a more general setting, for polynomials over a unique factorization domain (e.g. the ring of multivariate polynomials over a field or a PID).

Chinese remainder theorem secret sharing in multivariate polynomials

This paper deals with a generalization of the secret sharing using Chinese remainder theorem over the integers to multivariate polynomials over a finite field. We work with the ideals and their Gröbner bases instead of integer moduli. Therefore, the proposed method is called GB secret sharing. It was initially presented in our previous paper. Now we prove that any threshold structure has ideal GB realization. In a generic threshold modular scheme in ring of integers the sizes of the share space and the secret space are not equal. So, the scheme is not ideal and our generalization of modular secret sharing to the multivariate polynomial ring is more secure.

Fast Detection of Degenerate Predicates in Free Space Construction

An implementation of a computational geometry algorithm is robust if the combinatorial output is correct for every input. Robustness is achieved by ensuring that the predicates in the algorithm are evaluated correctly. A predicate is the sign of an algebraic expression whose variables are input parameters. The hardest case is detecting degenerate predicates where the value of the expression equals zero. We encounter this case in constructing the free space of a polyhedron that rotates around a fixed axis and translates freely relative to a stationary polyhedron. Each predicate involved in the construction is expressible as the sign of a univariate polynomial [Formula: see text] evaluated at a zero [Formula: see text] of a univariate polynomial [Formula: see text], where the coefficients of [Formula: see text] and [Formula: see text] are polynomials in the coordinates of the polyhedron vertices. A predicate is degenerate when [Formula: see text] is a zero of a common factor of [Formula: see text] and [Formula: see text]. We present an efficient degeneracy detection algorithm based on a one-time factoring of all the univariate polynomials over the ring of multivariate polynomials in the vertex coordinates. Our algorithm is 3500 times faster than the standard algorithm based on greatest common divisor computation. It reduces the share of degeneracy detection in our free space computations from 90% to 0.5% of the running time.

Standard Bases for Linear Codes over Prime Fields

It is known that a linear code can be represented by a binomial ideal. In this paper, we give standard bases for the ideals in a localization of the multivariate polynomial ring in the case of the linear codes over prime fields.

On ideal lattices, Gröbner bases and generalized hash functions

In this paper, we draw connections between ideal lattices and multivariate polynomial rings over integers using Gröbner bases. Univariate ideal lattices are ideals in the residue class ring, [Formula: see text] (here [Formula: see text] is a monic polynomial) and cryptographic primitives have been built based on these objects. Ideal lattices in the univariate case are generalizations of cyclic lattices. We introduce the notion of multivariate cyclic lattices and show that ideal lattices are a generalization of them in the multivariate case too. Based on multivariate ideal lattices, we construct hash functions using Gröbner basis techniques. We define a worst case problem, shortest substitution problem with respect to an ideal in [Formula: see text], and use its computational hardness to establish the collision resistance of the hash functions.

Polly Cracker, revisited

We formally treat cryptographic constructions based on the hardness of deciding ideal membership in multivariate polynomial rings. Of particular interest to us is a class of schemes known as "Polly...

Special issue on symbolic methods in multidimensional systems theory

In the seventies, the study of transfer matrices of linear time-invariant systems led to the development of the polynomial approach (Kailath, Rosenbrock, et al.). In this approach, univariate polynomial matrices play a central role (e.g., Hermite and Smith normal forms, Bezout equations etc.). When generalizing linear systems given by ordinary differential/difference equations to differential time-delay systems, systems with parameters, 2-D or 3-D filters and circuits, one had to face the case of systems described by means of matrices with entries in multivariate polynomial rings. These new systems were called 2-D or 3-D and, more generally, multidimensional systems (Bose, Lin, Pommaret, Oberst, Youla, Wood, et al.). For such systems, no normal forms such as Hermite and Smith forms exist. In order to handle these problems, the concept of Grobner bases (developed by Buchberger) was introduced in multidimensional systems theory. In many ways, the computation of these bases can be seen as an extension of both Gaussian elimination and the Euclidean algorithm. Grobner bases have been implemented in nearly all modern computer algebra systems. Another difficulty with multidimensional systems is the lack of a canonical first order representation. Several models (Roesser, Fornasini-Marchesini, reduced Spencer form etc.) have been studied extensively. The geometric approach to systems and control (Basile,Marro, Wonham, et al.) has recently been extended to these classes of multidimensional systems yielding promising results. The reduction to first order representations (realization problem) is of great importance to many applications of multidimensional systems theory such as uncertainty modeling for robust control. Special computer algebra packages have been developed for the effective manipulation of linear fractional transformations. In his pioneering work, Kalman developed a module-theoretic approach to the realization problem. Independently, the Japanese school of Sato, Kashiwara, and Kawai proposed an

A note on multivariate polynomial division and Gröbner bases

We first present purely combinatorial proofs of two facts: the well-known fact that a monomial ordering must be a well ordering, and the fact (obtained earlier by Buchberger, but not widely known) that the division procedure in the ring of multivariate polynomials over a field terminates even if the division term is not the leading term, but is freely chosen. The latter is then used to introduce a previously unnoted, seemingly weaker, criterion for an ideal basis to be Grobner, and to suggest a new heuristic approach to Grobner basis computations.

Reduced Gröbner bases and Macaulay–Buchberger Basis Theorem over Noetherian rings

In this paper, we extend the characterization of Z[x]/〈f〉, where f∈Z[x] to be a free Z-module to multivariate polynomial rings over any commutative Noetherian ring, A. The characterization allows us to extend the Gröbner basis method of computing a k-vector space basis of residue class polynomial rings over a field k (Macaulay–Buchberger Basis Theorem) to rings, i.e. A[x1,…,xn]/a, where a⊆A[x1,…,xn] is an ideal. We give some insights into the characterization for two special cases, when A=Z and A=k[θ1,…,θm]. As an application of this characterization, we show that the concept of Border bases can be extended to rings when the corresponding residue class ring is a finitely generated, free A-module.

Finite Fields, Gröbner Bases and Modular Secret Sharing

We propose a new application of the Gröbner basis methods. In particular, we construct generalized modular secret sharing. Basically, the secret sharing is the pair of splitting and restoring algorithms. Initially, the secret key is divided into the shares containing partial information of the secret so that only specified sets of participants can restore it pooling their subsets of shares together. We provide the implementation of both algorithms of secret sharing using Gröbner basis methods. Therefore, we call the proposed construction GB secret sharing. The most important feature of our approach is that the original modular construction in the ring of integers and polynomial ring Fq [x] over the Galois field Fq is generalized to the multivariate polynomial ring Fq [x 1, x 2, ‥, xn ]. We use the ideals defined by their Gröbner bases in the place of integer and polynomial moduli. First, we prove that any access structure possesses a GB realization in any multivariate polynomial ring Fq [x 1, x 2,‥, xn ]. This is the generalization of our previous result in the ring of polynomials over the Galois field Fq and the results of Mignotte, Asmuth and Bloom for the threshold access structures in the ring of integers. Second, we give the characterization and count the special zero-dimensional ideals in Fq [x 1, x 2,‥, xn ] which are necessary for the realization of GB secret sharing.

Microprocessor Implementation of Key Agreement Protocol Over the Ring of Multivariate Polynomials

Efficiency analysis of key agreement protocol with matrices over multivariate polynomial ring is presented. We consider protocol implementation for highly resource-restricted areas. Total number of simple bit operations requires computing common secret key is determined for different microprocessors types and estimated expected protocol run time. Comparison to traditional key agreement protocols (Diffie-Hellman and Elliptic Curve Diffie-Hellman) shows that key agreement protocol over multivariate polynomial ring is more suitable for areas where there are no specialized co-processors installed. Bibl. 12, tabl. 2 (in English; abstracts in English and Lithuanian).http://dx.doi.org/10.5755/j01.eee.116.10.893

Dynamical Gröbner bases over Dedekind rings

In this paper, we extend the notion of “dynamical Gröbner bases” introduced by the second author to Dedekind rings (with zero divisors). As an application, we dynamically solve the ideal membership problem and compute a generating set for the syzygy module over multivariate polynomial rings with coefficients in Dedekind rings. We also give a partial positive answer to a conjecture about Gröbner rings.