1,753 research outputs found

    Sample medium-term plans for mathematics

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    Finite Groebner bases in infinite dimensional polynomial rings and applications

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    We introduce the theory of monoidal Groebner bases, a concept which generalizes the familiar notion in a polynomial ring and allows for a description of Groebner bases of ideals that are stable under the action of a monoid. The main motivation for developing this theory is to prove finiteness theorems in commutative algebra and its applications. A major result of this type is that ideals in infinitely many indeterminates stable under the action of the symmetric group are finitely generated up to symmetry. We use this machinery to give new proofs of some classical finiteness theorems in algebraic statistics as well as a proof of the independent set conjecture of Hosten and the second author.Comment: 24 pages, adds references to work of Cohen, adds more details in Section

    Some NP-Hard Problems for the Simultaneous Coprimeness of Values of Linear Polynomials

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    The algorithmic-time complexity of some problems connected with linear polynomials and coprimeness relation on natural numbers is under consideration in the paper. We regard two easily stated problems. The first one is on the consistency in natural numbers from the interval of a linear coprimeness system. This problem is proved to be NP-complete. The second one is on the consistency in natural numbers of a linear coprimeness and discoprimeness system for polynomials with not greater than one non-zero coefficient. This problem is proved to be NP-hard. Then the complexity of some existential theories of natural numbers with coprimeness is considered. These theories are in some sense intermediate between the existential Presburger arithmetic and the existential Presburger arithmetic with divisibility. In a form of corollaries from the theorems of the second section we prove NP-hardness of the decision problem for the existential theories of natural numbers for coprimeness with addition and for coprimeness with successor function. In the conclusion section we give some remarks on the NP membership of the latter problem

    Scaled Boolean Algebras

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    Scaled Boolean algebras are a category of mathematical objects that arose from attempts to understand why the conventional rules of probability should hold when probabilities are construed, not as frequencies or proportions or the like, but rather as degrees of belief in uncertain propositions. This paper separates the study of these objects from that not-entirely-mathematical problem that motivated them. That motivating problem is explicated in the first section, and the application of scaled Boolean algebras to it is explained in the last section. The intermediate sections deal only with the mathematics. It is hoped that this isolation of the mathematics from the motivating problem makes the mathematics clearer.Comment: 53 pages, 8 Postscript figures, Uses ajour.sty from Academic Press, To appear in Advances in Applied Mathematic

    Richard Dedekind and the Creation of an Ideal: Early Developments in Ring Theory

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    Bounds on the Automata Size for Presburger Arithmetic

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    Automata provide a decision procedure for Presburger arithmetic. However, until now only crude lower and upper bounds were known on the sizes of the automata produced by this approach. In this paper, we prove an upper bound on the the number of states of the minimal deterministic automaton for a Presburger arithmetic formula. This bound depends on the length of the formula and the quantifiers occurring in the formula. The upper bound is established by comparing the automata for Presburger arithmetic formulas with the formulas produced by a quantifier elimination method. We also show that our bound is tight, even for nondeterministic automata. Moreover, we provide optimal automata constructions for linear equations and inequations

    Gaussian Integers and Dedekind\u27s Creation of an Ideal: A Number Theory Project

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