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4 results on '"[MATH.MATH-AC]Mathematics [math]/Commutative Algebra [math.AC]"'

1. MV-algebras and Partially Cyclically Ordered Groups

Author

Gérard Leloup, Laboratoire Manceau de Mathématiques (LMM), and Le Mans Université (UM)

Subjects
Pure mathematics, cyclically ordered abelian groups, [MATH.MATH-AC]Mathematics [math]/Commutative Algebra [math.AC], Existential quantification, partially cyclically ordered abelian groups, Cyclic group, Characterization (mathematics), Commutative Algebra (math.AC), 01 natural sciences, 010104 statistics & probability, Simple (abstract algebra), FOS: Mathematics, 0101 mathematics, Algebra over a field, Mathematics, MV-algebras, Algebra and Number Theory, [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA], 010102 general mathematics, Cyclic order, Mathematics - Logic, Mathematics - Rings and Algebras, Mathematics - Commutative Algebra, pseudofinite, MV-chains, [MATH.MATH-LO]Mathematics [math]/Logic [math.LO], Unimodular matrix, Computational Theory and Mathematics, Rings and Algebras (math.RA), Geometry and Topology, Logic (math.LO), Complex number
Abstract

We prove that there exists a functorial correspondence between MV-algebras and partially cyclically ordered groups which are wound round of lattice-ordered groups. It follows that some results about cyclically ordered groups can be stated in terms of MV-algebras. For example, the study of groups together with a cyclic order allows to get a first-order characterization of groups of unimodular complex numbers and of finite cyclic groups. We deduce a characterization of pseudofinite MV-chains and of pseudo-simple MV-chains (i.e. which share the same first-order properties as some simple ones). We can generalize these results to some non-lineraly ordered MV-algebras, for example hyper-archimedean MV-algebras.

Published
2021
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2. On differentially algebraic generating series for walks in the quarter plane

Author

Charlotte Hardouin, Michael F. Singer, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), ANR-19-CE40-0018,DeRerumNatura,Décider l'irrationalité et la transcendance(2019), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), and Department of Mathematics, North Carolina State University

Subjects
FOS: Computer and information sciences, Computer Science - Symbolic Computation, 05A15, 11G05, 30D05, 39A06, Polynomial, Pure mathematics, [MATH.MATH-AC]Mathematics [math]/Commutative Algebra [math.AC], General Mathematics, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], General Physics and Astronomy, Difference Galois theory, 0102 computer and information sciences, Symbolic Computation (cs.SC), 39A06 Lattice Walks, 01 natural sciences, Random Walks, [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], FOS: Mathematics, Mathematics - Combinatorics, Number Theory (math.NT), 0101 mathematics, Algebraic number, Mordell-Weil Lattices, October 2, Transcendence, Mathematics, Mathematics - Number Theory, Plane (geometry), Elliptic functions, 010102 general mathematics, 2020. 2010 Mathematics Subject Classification. 05A15, Elliptic function, Kodaira-Néron Model, Random walk, Generating Functions, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], 30D05, Néron–Tate height, 010201 computation theory & mathematics, Generating series, Combinatorics (math.CO), [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], 11G05, Néron-Tate Height
Abstract

We refine necessary and sufficient conditions for the generating series of a weighted model of a quarter plane walk to be differentially algebraic. In addition, we give algorithms based on the theory of Mordell-Weil lattices, that, for each weighted model, yield polynomial conditions on the weights determining this property of the associated generating series., 37 Pages

Published
2021
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3. On the Discriminant Scheme of Homogeneous Polynomials

Author

Laurent Busé, Jean-Pierre Jouanolou, Géométrie , Algèbre, Algorithmes (GALAAD2), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut de Recherche Mathématique Avancée (IRMA), and Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
[INFO.INFO-SC]Computer Science [cs]/Symbolic Computation [cs.SC], Power sum symmetric polynomial, [MATH.MATH-AC]Mathematics [math]/Commutative Algebra [math.AC], Applied Mathematics, inertia forms, Complete homogeneous symmetric polynomial, resultant of homogeneous polynomials, Classical orthogonal polynomials, Combinatorics, Computational Mathematics, Elimination theory, Computational Theory and Mathematics, Discriminant, Difference polynomials, Homogeneous polynomial, discriminant of homogeneous polynomials, Orthogonal polynomials, Elementary symmetric polynomial, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], Mathematics
Abstract

In this paper, the discriminant scheme of homogeneous polynomials is studied in two particular cases: the case of a single homogeneous polynomial and the case of a collection of n − 1 homogeneous polynomials in $${n\geqslant 2}$$ variables. In both situations, a normalized discriminant polynomial is defined over an arbitrary commutative ring of coefficients by means of the resultant theory. An extensive formalism for this discriminant is then developed, including many new properties and computational rules. Finally, it is shown that this discriminant polynomial is faithful to the geometry: it is a defining equation of the discriminant scheme over a general coefficient ring k, typically a domain, if $${2\neq 0}$$ in k. The case where 2 = 0 in k is also analyzed in detail.

Published
2014
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4. On the Modular Computation of Gröbner Bases with Integer Coefficients

Author

S. Yu. Orevkov, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, Discrete mathematics, Ring (mathematics), Sequence, Mathematics::Commutative Algebra, Markov chain, [MATH.MATH-AC]Mathematics [math]/Commutative Algebra [math.AC], Applied Mathematics, General Mathematics, Computation, 010102 general mathematics, 010103 numerical & computational mathematics, 16. Peace & justice, Base (topology), 01 natural sciences, Combinatorics, Integer, Ideal (ring theory), [MATH]Mathematics [math], 0101 mathematics, ComputingMilieux_MISCELLANEOUS, Monomial order, Mathematics
Abstract

Let I1 ⊂ I2 ⊂ . . . be an increasing sequence of ideals of the ring Z[X], X = (x1, . . . , xn), and let I be their union. We propose an algorithm to compute the Gr¨obner base of I under the assumption that the Gr¨obner bases of the ideal QI of the ring Q[X] and of the ideals I ⊗ (Z/mZ) of the rings (Z/mZ)[X] are known. Such an algorithmic problem arises, for example, in the construction of Markov and semi-Markov traces on cubic Hecke algebras.

Published
2014
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