Contemporary Developments in Finite Fields and Applications, pp. 296-320 (2016) No AccessFinding a Gröbner basis for the ideal of recurrence relations on m-dimensional periodic arraysIvelisse M. Rubio, Moss Sweedler and Chris HeegardIvelisse M. RubioDepartment of Computer Science, University of Puerto Rico, Box 70377, S.J., PR 00936-8377, Puerto Rico, Moss SweedlerDepartment of Mathematics, Cornell University, 310 Malott Hall, Ithaca, NY 14853, USA and Chris HeegardNative Intelligence, 17179 La Brisa Ct., Sugarloaf, FL 33042, USAhttps://doi.org/10.1142/9789814719261_0018Cited by:4 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail Abstract: Recent developments in applications of multidimensional periodic arrays [9] have drawn new attention to the computation of Gröbner bases for the ideal of linear recurrence relations on the arrays. An m-dimensional infinite array can be represented by a multivariate power series sitting within the ring of multivariate Laurent series. We reinterpret the problem of finding linear recurrence relations on m-dimensional periodic arrays as finding the kernel of a module map involving quotients of Laurent series and present an algorithm to compute a Gröbner basis for this kernel. The algorithm does not assume the knowledge of a generating set for the kernel of this ideal and it is based on linear algebra computations. Finding a generating set is one application of the algorithm. FiguresReferencesRelatedDetailsCited By 4Decoding up to 4 Errors in Hyperbolic-Like Abelian Codes by the Sakata AlgorithmJosé Joaquín Bernal and Juan Jacobo Simón17 February 2021A New Approach to the Berlekamp-Massey-Sakata Algorithm: Improving Locator DecodingJose Joaquin Bernal-Buitrago and Juan Jacobo Simon-Pinero1 Jan 2021 | IEEE Transactions on Information Theory, Vol. 67, No. 1Multidimensional linear complexity analysis of periodic arraysRafael Arce-Nazario, Francis Castro, Domingo Gomez-Perez, Oscar Moreno and José Ortiz-Ubarri et al.5 July 2019 | Applicable Algebra in Engineering, Communication and Computing, Vol. 31, No. 1Arrays composed from the extended rational cycleDomingo Gomez-Perez, Ana-Isabel Gomez and Andrew Tirkel1 Jan 2017 | Advances in Mathematics of Communications, Vol. 11, No. 2 Contemporary Developments in Finite Fields and ApplicationsMetrics History PDF download

Let H denote an algebra of input symbols or events. If X is the state space for a system, then one can form the space R of observations of X. Under suitable conditions, both X and R are H-modules. Loosely speaking, the formal systems studied in this paper consists of a bialgebra H describing the input symbols and two H-modules describing the states and observations of the system. Finite automata and input-output systems are concerned with commutative R, while quantum systems, such as quantum automata, are concerned with non-commutative R arising from Hermitian operators on the state space. Of special interest are those systems consisting of interacting networks of classical systems and automata. These types of systems have become known as hybrid systems and are examples of formal systems with commutative R. In this paper, we present a number of examples of hybrid systems and quantum automata and point out some relationships between them. This is a preliminary announcement: a detailed exposition, including proofs, will appear elsewhere.

Classical Buchberger theory is generalized to a new family of rings. The family includes all subalgebras of the polynomial algebra in one variable. Some subalgebras of polynomial algebras in several variables are included. The new rings are integral domains and have a number of other properties in common with polynomial rings. The rings sit in a field F in appropriate position relative to a valuation ring in F . The valuation ring gives rise to a totally ordered monoid which plays the role of a term ordering in the classical Buchberger theory. The paper is reasonably self-contained and contains an infinite number of examples.

This article analyses the importance of the writings of Laure (pseudonym of Colette Peignot) for Georges Bataille's communitarian projects of the late 1930s and early 1940s and for Maurice Blanchot's interpretation of these projects. Through readings of theoretical essays by Bataille, his annotated edition of Laure's Le Sacre (published illegally in 1939, only months after the author's death, and distributed clandestinely to a restricted group of readers), and Blanchot's La Communaute inavouable, I argue that Laure's book functions as the literary analogue to two avant-garde communities in which Laure and Bataille were involved - Acephale and the College of Sociology - and that this same book is at the heart of the unavowable community theorized by Blanchot.Bataille's justification for publishing Laure's writings echoes the very terms he uses to describe his group projects of the late 1930s: community, communication, the sacred, sacrifice. I argue that, in publishing and distributing her book, Bataille effectively sanctifies Laure, turning her into a martyr for the community. However, in contrast to Acephale and the College of Sociology, the community founded over Laure's dead body was a virtual community: one based on the members' solitary experiences of reading. Laure becomes the figurehead of an unavowable community, as Blanchot's locution would have it.

Suppose I is a prime ideal in k[X1, ...,Xn] with a given finite generating set and k(q1,...,qm) is a finitely generated subfield of the field of fractions Z of k[X1, ..., Xn]/I and c is an element of Z. We present Groebner basis techniques to determine: The information about c also tells whether c lies in k(q1,...,qm), solving the subfield membership problem. Determination of the algebraic or transcendental nature of Z over k(q1,...,qm) includes finding the index in case of algebraicity or transcendence degree in case the extension is transcendental. The determination of the nature of Z over k(q1,...,qm) is not simply an iterative application of the results for c and only requires computing one Groebner basis.

Given a valuation on the function field k( x; y), we examine the set of images of nonzero elements of the underlying polynomial ring k[ x; y] under this valuation. For an arbitrary field k, a Noetherian power series is a map z : Q --> k that has Noetherian (i.e., reverse well-ordered) support. Each Noetherian power series induces a natural valuation on k( x; y). Although the value groups corresponding to such valuations are well-understood, the restrictions of the valuations to underlying polynomial rings have yet to be characterized. Let Lambda(n) denote the images under the valuation v of all nonzero polynomials f is an element of k[ x; y] of at most degree n in the variable y. We construct a bound for the growth of Lambda(n) with respect to n for arbitrary valuations, and then specialize to valuations that arise from Noetherian power series. We provide a sufficient condition for this bound to be tight.

The classical theory of Gröbner bases, as developed by Bruno Buchberger, can be expanded to utilize objects more general than term orders. Each term order on the polynomial ring k [x] produces a filtration of k [x] and a valuation ring of the rational function fieldk (x). The algorithms developed by Buchberger can be performed by using directly the induced valuation or filtration in place of the term order. There are many valuations and filtrations that are suitable for this general computational framework that are not derived from term orders, even after a change of variables. Here we study how to translate between properties of filtrations and properties in valuation theory, and give a characterization of which valuations and filtrations are derived from a term order after a change of variables. This characterization illuminates the properties of valuations and filtrations that are desirable for use in a generalized Gröbner basis theory.

Abstract: Given a (skew) valuation on a multivariate functioneld k(x), weexamine the set of images of nonzero elements of an underlying polynomialring k[x] under this valuation. For an arbitraryeld k, a Noetherian powerseries is a map z : Q ! k that has Noetherian (i.e., reverse well-ordered)support. The study of such series leads to many natural valuations on functionelds, and this becomes one of our points of focus. Although the value groupscorresponding to such valuations are...

The study of Grobner bases requires the use of term orders to perform various algorithms involving multivariate polynomials. Using techniques developed by Moss Sweedler and Lorenzo Robbiano, it is possible to perform similar computations in a more general setting. In this manuscript, we give an exposition of the general theory of Grobner bases and demonstrate how this theory specializes to the classical case in which term orders are utilized. Given a polynomial ring R, we define a general filtration to be a nested sequence of subsets of R. We can form a correspondence between valuations on the rational function field k( x1,…,xn) and filtrations on k[x1,…, xn] with special properties. The fundamental concept which allows Buchberger's classical reduction algorithms to be performed in finite time is the well-ordered property of term orders. In the spirit of the work laid out by Moss Sweedler and Lorenzo Robbiano, we consider methods of polynomial computations without the direct use of term orders. Instead, Grobner bases are constructed by using valuations having special properties. Although we present our computational theory in the framework of valuations, we could have just as well developed our theory in terms of valuation rings or filtrations with special additional properties. In this spirit, we formulate and discuss a correspondence between filtrations, valuations, and valuation rings in the intermediate chapters of the manuscript. In light of this correspondence, we give a criterion for the case when filtrations, valuations, and valuation rings arise from a term order in suitable variables. This characterization leads us to search for other well-ordered valuations which do not come from a term order in suitable variables. We close by giving the first examples of well-ordered, zero-dimensional valuations on k(x, y) with respect to the underlying polynomial ring k[x, y]. An infinite family of such valuations where k is a field of characteristic zero is given. Additionally, we give characteristic-free examples of well-ordered valuations, and in positive characteristic, we construct valuations which are not well-ordered.

This note is about a methodology utilizing inexact computation in conjunction with exact computation where the exact input is known and exact output is desired. The inexact computation is used to help avert the growth of intermediate expressions. This growth frequently makes using exact computation infeasible. We mention several existing applications and also mention where the methodology is not useful. We propose new directions where one can make effective use of the stabilization methodology.

For an ideal or K K -subalgebra E E of K [ X 1 , … , X n ] K[X_1,\dots ,X_n] , consider subfields k ⊂ K k\subset K , where E E is generated – as ideal or K K -subalgebra – by polynomials in k [ X 1 , … , X n ] k[X_1,\dots ,X_n] . It is a standard result for ideals that there is a smallest such k k . We give an algorithm to find it. We also prove that there is a smallest such k k for K K -subalgebras. The ideal results use reduced Gröbner bases. For the subalgebra results we develop and then use subduced SAGBI (bases), the analog to reduced Gröbner bases.

We present a new algorithm for finding linear recursion relations using Grobner bases. The algorithm is applied to the decoding of hyperbolic cascaded Reed-Solomon codes and algebraic geometric codes

From the Publisher: Anil Nerode has had a wide influence on logic and computer science since the 1960's. This volume reflects his inspiration and also the variety of interests which he has. The papers in this book are principally concerned with mathematical logic and some of its applications in computing. The book contains papers on recursion theory, intuitionism, computability in group theory, recursive model theory, reverse mathematics, and the extraction of programs from proofs. Included also is a thorough survey of Nerode's technical achievements over the last 30 years. Any logician should find something, and most probably many items, of interest. We note particularly new developments in the understanding of the property of intuitionistic set theory and intuitionistic analysis, the extension of computability in ordinary mathematics pioneered by Marian Boyka Pour-El and Ian Richards, the proof-theoretic strength of a long-standing conjecture of Fraisse, and an extension and further account of the Curry-Howard method of extracting programs from logical proofs. The papers in general arose from the conference Logical Methods in Mathematics and Computer Science. A Symposium in Honor of Anil Nerode on the Occasion of his Sixtieth Birthday at the Mathematical Sciences Institute, Cornell University, from June 1-3, 1992. This conference was attended by over 100 participants and reflects the range and influence of Anil's work.

For given f1,…,fm ϵ K[x] which are relatively prime we present degree bounds on the ai needed to express 1 and other "low degree" polynomials as ∑aifi. This paper gives an improvement on Kakié's bound (Kakié, 1976).

k is a field, X"1,..., X"n are indeterminates over k and f"1,...,f"m@?k[X"1,...,X"n]. This note presents a simple algorithm, based on Grobner bases, to test if a given polynomial g of k[X"1,...,X"n] lies in k[f"1,...,f"m]. If so, the algorithm produces a polynomial P of m variables where g=P(f"1,...,f"m). Say @w: B->k[X"1,...,X"n] is a homomorphism where @w(b"i)=f"i, for algebra generators {b"i}@?B. If @w is onto, our algorithm gives a homomorphism @l:k[X"1,...,X"n]->B, where the composite @w@l is the identity map. In particular, the algorithm computes the inverse of algebra automorphisms of the polynomial ring. A variation of our test if k[f"1,...,f"m]=k[X"1,...,X"n], tells if k(f"1,...,f"m)=k(X"1,...,X"n). Existing computer algebra systems, such as IBM's SCRATCHPAD II, have Grobner basis packages which allow the user to specify a term ordering sufficient to carry out the algorithm.

The Hilbert generating function of TorU(A, A) gives the inverse of the Hilbert generating function of U where U is a graded A algebra.