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PRESBURGER ARITHMETIC, RATIONAL GENERATING FUNCTIONS, AND QUASIPOLYNOMIALS
"... Presburger arithmetic is the firstorder theory of the natural numbers with addition (but no multiplication). We characterize sets that can be defined by a Presburger formula as exactly the sets whose characteristic functions can be represented by rational generating functions; a geometric characte ..."
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Cited by 3 (2 self)
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as, equivalently, either a piecewise quasipolynomial or a rational generating function. Finally, we translate known computational complexity results into this setting and discuss open directions. §1. Introduction. A broad and interesting class of sets are those that can be defined over N = {0, 1, 2
The unreasonable ubiquitousness of quasipolynomials
, 2013
"... A function g, with domain the natural numbers, is a quasipolynomial if there exists a period m and polynomials p0, p1,..., pm−1 such that g(t) = pi(t) for t ≡ i mod m. Quasipolynomials classically – and “reasonably ” – appear in Ehrhart theory and in other contexts where one examines a family of ..."
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A function g, with domain the natural numbers, is a quasipolynomial if there exists a period m and polynomials p0, p1,..., pm−1 such that g(t) = pi(t) for t ≡ i mod m. Quasipolynomials classically – and “reasonably ” – appear in Ehrhart theory and in other contexts where one examines a family
Quasipolynomials and the Bethe Ansatz
, 2008
"... We study solutions of the Bethe Ansatz equation related to the trigonometric Gaudin model associated to a simple Lie algebra g and a tensor product of irreducible finitedimensional representations. Having one solution, we describe a construction of new solutions. The collection of all solutions obt ..."
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operators. The dynamical Weyl group projectively acts on the common eigenvectors of the trigonometric Gaudin operators. We conjecture that this action preserves the set of Bethe vectors and coincides with the action induced by the action on points of populations. We prove the conjecture for sl2.
Maximal periods of (Ehrhart) quasipolynomials
 the electronic journal of combinatorics 17 (2010), #R68 12 Benjamin Braun, Norm
, 2008
"... Abstract. A quasipolynomial is a function defined of the form q(k) = cd(k) k d + cd−1(k)k d−1 + · · · + c0(k), where c0, c1,..., cd are periodic functions in k ∈ Z. Prominent examples of quasipolynomials appear in Ehrhart’s theory as integerpoint counting functions for rational polytopes, and M ..."
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Cited by 10 (2 self)
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Abstract. A quasipolynomial is a function defined of the form q(k) = cd(k) k d + cd−1(k)k d−1 + · · · + c0(k), where c0, c1,..., cd are periodic functions in k ∈ Z. Prominent examples of quasipolynomials appear in Ehrhart’s theory as integerpoint counting functions for rational polytopes
Computing the period of an Ehrhart quasipolynomial
 Electron. J. Combin., 12:Research Paper
, 2005
"... If P ⊂ R d is a rational polytope, then iP(t): = #(tP ∩ Z d) is a quasipolynomial in t, called the Ehrhart quasipolynomial of P. A period of iP(t) is D(P), the smallest D ∈ Z+ such that D · P has integral vertices. Often, D(P) is the minimum period of iP(t), but, in several interesting examples, t ..."
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Cited by 6 (0 self)
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If P ⊂ R d is a rational polytope, then iP(t): = #(tP ∩ Z d) is a quasipolynomial in t, called the Ehrhart quasipolynomial of P. A period of iP(t) is D(P), the smallest D ∈ Z+ such that D · P has integral vertices. Often, D(P) is the minimum period of iP(t), but, in several interesting examples
The irreducibility of the space of curves of given genus
 Publ. Math. IHES
, 1969
"... Fix an algebraically closed field k. Let Mg be the moduli space of curves of genus g over k. The main result of this note is that Mg is irreducible for every k. Of course, whether or not M s is irreducible depends only on the characteristic of k. When the characteristic s o, we can assume that k ~ ..."
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Cited by 512 (2 self)
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from char. o to char. p provided that p> 2g qi. Unfortunately, attempts to extend this method to all p seem to get stuck on difficult questions of wild ramification. Nowadays, the Teichmtiller theory gives a thoroughly analytic but very profound insight into this irreducibility when kC. Our
Coefficient functions of the Ehrhart quasipolynomials of rational polygons
, 906
"... polynomials of convex integral polygons. We study the same question for Ehrhart polynomials and quasipolynomials of nonintegral convex polygons. Define a pseudointegral polygon, or PIP, to be a convex rational polygon whose Ehrhart quasipolynomial is a polynomial. The numbers of lattice points on ..."
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Cited by 2 (1 self)
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remains open. Turning to the case in which the Ehrhart quasipolynomial has nontrivial quasiperiod, we determine the possible minimal periods that the coefficient functions of the Ehrhart quasipolynomial of a rational polygon may have.
On the Ring of Integervalued Quasipolynomial
"... The paper studies some properties of the ring of integervalued quasipolynomials. On this ring, theory of generalized Euclidean division and generalized greatest common divisor are presented. Applications to finite simple continued fraction expansion of rational numbers and Smith normal form of int ..."
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The paper studies some properties of the ring of integervalued quasipolynomials. On this ring, theory of generalized Euclidean division and generalized greatest common divisor are presented. Applications to finite simple continued fraction expansion of rational numbers and Smith normal form
Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization
 SIAM Journal on Optimization
, 1993
"... We study the semidefinite programming problem (SDP), i.e the problem of optimization of a linear function of a symmetric matrix subject to linear equality constraints and the additional condition that the matrix be positive semidefinite. First we review the classical cone duality as specialized to S ..."
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Cited by 557 (12 self)
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We study the semidefinite programming problem (SDP), i.e the problem of optimization of a linear function of a symmetric matrix subject to linear equality constraints and the additional condition that the matrix be positive semidefinite. First we review the classical cone duality as specialized
Results 1  10
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27,186