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12
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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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
, 2006
"... 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. ..."
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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.
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 ..."
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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
Model Theory and Differential Algebra
, 2001
"... This paper stems from my lecture notes for a talk given at Rutgers University in Newark on 3 November 2000 as part of the Workshop on Dierential Algebra and Related Topics. I thank Li Guo for inviting me to speak and for organizing such a successful meeting of the disparate strands of the dierential ..."
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This paper stems from my lecture notes for a talk given at Rutgers University in Newark on 3 November 2000 as part of the Workshop on Dierential Algebra and Related Topics. I thank Li Guo for inviting me to speak and for organizing such a successful meeting of the disparate strands of the dierential algebra community. I thank also the referee for carefully reading an earlier version of this note and for suggesting many improvements
HFields And Their Liouville Extensions
"... . We introduce Helds as ordered dierential elds of a certain kind. Hardy elds extending R, as well as the eld of logarithmicexponential series over R are Helds. We study Liouville extensions in the category of Helds, as a step towards a model theory of Helds. The main result is that an Hel ..."
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. We introduce Helds as ordered dierential elds of a certain kind. Hardy elds extending R, as well as the eld of logarithmicexponential series over R are Helds. We study Liouville extensions in the category of Helds, as a step towards a model theory of Helds. The main result is that an Held has at most two Liouville closures. Contents Introduction 1 1. HFields and DierentialValued Fields 4 2. Asymptotic Couples 6 3. Algebraic Extensions of HFields 15 4. Embedding PreHFields into HFields 19 5. Simple Transcendental Liouville Extensions 25 6. Liouville Closures 30 7. Concluding Remarks 35 References 36 Introduction There are two algebraically avoured theories about the asymptotic behaviour at innity of real valued functions on halines (a; +1), with a 2 R. One is the theory of Hardy elds (see Bourbaki [5], Rosenlicht [18], [19], [20]). The other concerns the eld of \transseries" ( Ecalle [7], Van der Hoeven [9]), or in an essentially equivalent termin...
Differentially Algebraic Gaps
, 2003
"... Hfields are ordered differential fields that capture some basic properties of Hardy fields and fields of transseries. Each Hfield is equipped with a convex valuation, and solving firstorder linear differential equations in Hfield extensions is strongly affected by the presence of a "gap& ..."
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Hfields are ordered differential fields that capture some basic properties of Hardy fields and fields of transseries. Each Hfield is equipped with a convex valuation, and solving firstorder linear differential equations in Hfield extensions is strongly affected by the presence of a "gap" in the value group. We construct a real closed Hfield that solves every firstorder linear differential equation, and that has a differentially algebraic Hfield extension with a gap. This answers a question raised in [1]. The key is a combinatorial fact about the support of transseries obtained from iterated logarithms by algebraic operations, integration, and exponentiation.
Towards a Model Theory for Transseries
"... 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 o ..."
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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.