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ORDERINGS OF MONOMIAL IDEALS
, 2003
"... We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering given by reverse inclusion. We give a short proof of the fact that every antichain of monomial ideals is finite. Then we investigate ordinal invariants for the complexity of this ordered set. In particular ..."
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Cited by 4 (1 self)
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We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering given by reverse inclusion. We give a short proof of the fact that every antichain of monomial ideals is finite. Then we investigate ordinal invariants for the complexity of this ordered set. In particular, we give an interpretation of the height function in terms of the HilbertSamuel polynomial, and we compute upper and lower bounds on the maximal order type.
Finiteness theorems in stochastic integer programming
 Foundations of Computational Mathematics
, 2003
"... Abstract. We study Graver test sets for families of linear multistage stochastic integer programs with varying number of scenarios. We show that these test sets can be decomposed into finitely many “building blocks”, independent of the number of scenarios, and we give an effective procedure to comp ..."
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Cited by 3 (0 self)
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Abstract. We study Graver test sets for families of linear multistage stochastic integer programs with varying number of scenarios. We show that these test sets can be decomposed into finitely many “building blocks”, independent of the number of scenarios, and we give an effective procedure to compute them. The paper includes an introduction to NashWilliams ’ theory of betterquasiorderings, which is used to show termination of our algorithm. We also apply this theory to finiteness results for Hilbert functions.
Complexity Bounds for ZeroTest Algorithms
, 2001
"... In this paper, we analyze the complexity of a zero test for expressions built from formal power series solutions of first order differential equations with non degenerate initial conditions. We will prove a doubly exponential complexity bound. This bound establishes a power series analogue for "witn ..."
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In this paper, we analyze the complexity of a zero test for expressions built from formal power series solutions of first order differential equations with non degenerate initial conditions. We will prove a doubly exponential complexity bound. This bound establishes a power series analogue for "witness conjectures".
Liouville closed Hfields
 J. Pure Appl. Algebra
"... Abstract. Hfields are fields with an ordering and a derivation subject to some compatibilities. (Hardy fields extending R and fields of transseries over R are Hfields.) We prove basic facts about the location of zeros of differential polynomials in Liouville closed Hfields, and study various cons ..."
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Abstract. Hfields are fields with an ordering and a derivation subject to some compatibilities. (Hardy fields extending R and fields of transseries over R are Hfields.) We prove basic facts about the location of zeros of differential polynomials in Liouville closed Hfields, and study various constructions in the category of Hfields: closure under powers, constant field extension, completion, and building Hfields with prescribed constant field and Hcouple. We indicate difficulties in obtaining a good model theory of Hfields, including an undecidability
ZeroTesting, Witness Conjectures and Differential Diophantine Approximation
, 2001
"... Consider a class of constants built up from the rationals using the field operations and a certain number of transcendental functions like exp. A central problem in computer algebra is to test whether such a constant, which is represented by an expression, is zero. The simplest approach to the zero ..."
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Consider a class of constants built up from the rationals using the field operations and a certain number of transcendental functions like exp. A central problem in computer algebra is to test whether such a constant, which is represented by an expression, is zero. The simplest approach to the zerotest problem ist to evaluate...
Towards a Model Theory for Transseries Matthias Aschenbrenner, Lou
"... Abstract The differential field of transseries extends the field of real Laurent series, and occurs in various context: asymptotic expansions, analytic vector fields, ominimal structures, to name a few. We give an overview of the algebraic and modeltheoretic aspects of this differential field, and ..."
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Abstract The differential field of transseries extends the field of real Laurent series, and occurs in various context: asymptotic expansions, analytic vector fields, ominimal structures, to name a few. We give an overview of the algebraic and modeltheoretic aspects of this differential field, and report on our efforts to understand its firstorder theory.