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Topic: Ehrhart polynomial

	Ehrhart polynomial
		Ehrhart showed in 1967 that L is a rational polynomial of degree n in t i.e.
		Formulas for the other coefficients are harder to get; Todd classes of toric the Riemann-Roch theorem as well as Fourier analysis have been used for this purpose.
		The Ehrhart polynomial of the interior of a closed polytope P can be computed as:
	www.freeglossary.com /Ehrhart_polynomials (735 words)

	Learn more about Polynomial in the online encyclopedia. (Site not responding. Last check: 2007-10-31)
		The culmination of these efforts is Taylor's theorem, which roughly states that every differentiable function locally looks like a polynomial, and the Weierstrass approximation theorem, which states that every continuous function defined on a compact interval of the real axis can be approximated on the whole interval as closely as desired by a polynomial.
		Depending on the degree of the polynomial to be considered, simply checking if the polynomial has linear factors can eliminate several cases, and then resorting to checking divisibility of some other irreducible polynomials, however Eisenstein's criterion can be used to more efficiently determine irreducibility.
		Ehrhart polynomial (It is appropriate that this title is singular although some of the other special polynomials named after persons that are listed here are plural, because those are special polynomial sequences.)
	www.onlineencyclopedia.org /p/po/polynomial.html (1534 words)

	Wikipedia: Ehrhart polynomial
		Ehrhart showed in 1967 that L is a rational polynomial of degree n in t, i.e.
		Formulas for the other coefficients are much harder to get; Todd classes of toric varieties, the Riemann-Roch theorem as well as Fourier analysis have been used for this purpose.
		The Ehrhart polynomial of the interior of a closed polytope P can be computed as
	www.factbook.org /wikipedia/en/e/eh/ehrhart_polynomial.html (298 words)

	Ehrhart polynomial - Wikipedia, the free encyclopedia
		In mathematics, integral polytopes have associated Ehrhart polynomials which encode some geometrical information about them.
		Ehrhart showed in 1962 that L is a rational polynomial of degree n in t, i.e.
		Formulas for the other coefficients are much harder to get; Todd classes of toric varieties, the Riemann–Roch theorem as well as Fourier analysis have been used for this purpose.
	en.wikipedia.org /wiki/Ehrhart_polynomial (308 words)

	ipedia.com: Polynomial Article (Site not responding. Last check: 2007-10-31)
		In algebra, a polynomial function, or polynomial for short, is a function of the form where x is a scalar -valued variable, n is a nonnegative integer, and a 0,..., a n are fixed scalars, called the c...
		The culmination of these efforts is Taylor's theorem, which roughly states that every differentiable function locally looks like a polynomial, and the Stone-Weierstrass theorem, which states that every continuous function defined on a compact interval of the real axis can be approximated on the whole interval as closely as desired by a polynomial.
		Polynomials are also frequently used to interpolate functions.
	www.ipedia.com /polynomial.html (1572 words)

	Ehrhart polynomial: Facts and details from Encyclopedia Topic (Site not responding. Last check: 2007-10-31)
		Ehrhart showed in 1967 that L is a rational polynomial[Click link for more facts about this topic] of degree n in t, EHandler: no quick summary.
		(the bernoulli polynomials occur in the study of many special functions and in particular the riemann zeta function and the hurwitz zeta...
		The minimal polynomial of an n-by-n matrix a over a field (mathematics)field f is the monic polynomial p(x) over f of least...
	www.absoluteastronomy.com /encyclopedia/e/eh/ehrhart_polynomial.htm (1037 words)

	DMV - Fachgruppe Diskrete Mathematik
		The Ehrhart polynomial of a convex lattice polytope P counts the number of integer points in integral dilates of P. We will discuss what information about P can be obtained by expanding the Ehrhart polynomial in different bases, and present new linear inequalities satisfied by the coefficients of Ehrhart polynomials.
		We show that faster and simpler polynomial time approximation schemes for several scheduling problems can be obtained by reducing the number of jobs to a constant (that depends on the desired approximation error), and applying enumeration or dynamic programming afterwards.
		The chromatic polynomial P(G,k) of a graph G is the number of vertex-colorings of G with k available colors.
	www.ti.inf.ethz.ch /dm/radopreis2004.html (1209 words)

	[No title]
		We obtain closed forms in terms of cotangent expansions for the coefficients of the Ehrhart polynomial, that shed additional light on previous descriptions of the Ehrhart polynomial.
		Algebraic geometers have shown that the Hilbert polynomials of toric varieties associated to convex lattice polytopes precisely describe the number of lattice points inside their dilates \cite{3}.
		\end{equation} Ehrhart* \cite{4} inaugurated the systematic study of general properties of this function by proving that it is always a polynomial in $t \in \mathbb{N}$, and that in fact \begin{equation}\latticeP = \text{Vol} (\Pe) t^{n} + \frac{1}{2} \text{Vol} (\partial \Pe) t^{n-1} + \cdots + \chi (\Pe) \end{equation} for closed polytopes $\Pe $.
	www.emis.de /journals/ERA-AMS/1996-01-001/1996-01-001.tex.html (686 words)

	Stating and Manipulating Periodicity in the Polytope Model. Applications to Program Analysis and Optimization.
		The main contribution of this thesis is an enhancement of this model, by integrating the periodic character of extremal integer points of the Z-polyhedron, using "periodic polyhedra".
		Using this, we derive a new condition for which the Ehrhart polynomial of a Z-polyhedron is non-periodic.
		In one of the two cases, we also give a new method for computing this polynomial, whose complexity is polynomial for a fixed dimension.
	www.hipeac.net /node/57/print (362 words)

	Talks in 2004 (Site not responding. Last check: 2007-10-31)
		In particular, multivariate polynomial functions arise in many situations resulting from linearized subscripts in array reference functions, or from induction variable recognition, and compilers fail in handling such expressions.
		Bernstein polynomials are particular polynomials that form a basis for the space of polynomials.
		Due to the Bernstein convex hull property, the value of the polynomial is then bounded by the values of the minimum and maximum Bernstein coefficients.
	www.ifi.uni-klu.ac.at /Colloquia/1998/1998/2004/2702Clauss?pp=1& (480 words)

	New Ehrhart (quasi-)polynomials
		, we have calculated the Ehrhart polynomials and quasi-polynomials for polytopes that are slices or nice truncations of the unit
		Proposition 8 The Ehrhart quasi-polynomial for the truncated unit cube in Figure 4, where its vertices are at
		Proposition 9 The Ehrhart quasi-polynomial for the cuboctahedron (Figure 5) is:
	www.math.ucdavis.edu /~latte/theory/lattE/node6.html (276 words)

	Polynomial (Site not responding. Last check: 2007-10-31)
		Polynomial of fifth degree may be computed with four multiplications and five additions, and a
		Polynomial of sixth degree may be computed with four multiplications and seven additions.
		Polynomials are not soluble in the same manner.
	www.math.sdu.edu.cn /mathency/math/p/p480.htm (284 words)

	Ehrhart polynomial: Encyclopedia topic (Site not responding. Last check: 2007-10-31)
		Specifically, consider a lattice (lattice: Framework consisting of an ornamental design made of strips of wood or metal) L in Euclidean space (Euclidean space: A space in which Euclid's axioms and definitions apply; a metric space that is linear and finite-dimensional) R
		Ehrhart showed in 1967 that L is a rational polynomial (polynomial: A mathematical expression that is the sum of a number of terms) of degree n in t, i.e.
		The Ehrhart polynomial of the interior (interior: The region that is inside of something) of a closed polytope P can be computed as:
	www.absoluteastronomy.com /reference/ehrhart_polynomial (556 words)

	Department of Mathematics at MIT \| Graduate Study : Fall 2005 Thesis Defenses (Site not responding. Last check: 2007-10-31)
		In the 1960's Eugene Ehrhart discovered that for any rational d-polytope P; the number of lattice points, i(P,m), in the mth dilated polytope mP is always a quasi-polynomial of degree d in m; whose period divides the least common multiple of the denominators of the coordinates of the vertices of P.
		In the first part of my thesis, motivated by a conjecture given by De Loera, which gives a simple formula of the Ehrhart polynomial of an integral cyclic polytope, we define a more general family of polytopes, lattice-face polytopes, and show that these polytopes have the same simple form of Ehrhart polynomials.
		No polynomial algorithm is known to compute the best BST for a given sequence of key accesses, and before our work, no sub O(log(n))-competitive online BST data structures were known to exist.
	www-math.mit.edu /text-only/graduate/thesis-defense.html (2827 words)

	The Ehrhart polynomial of the Birkhoff polytope
		We present a new, complex-analytic way to compute the Ehrhart polynomial of the Birkhoff polytope, that is, the function counting the integer points in the dilated polytope.
		The leading term of the Ehrhart polynomial is--up to a trivial factor--the volume of the polytope, which is one reason why we are interested in this counting function.
		We implemented our methods in the form of a computer program, which yielded the Ehrhart polynomial (and hence the volume) of the ninth Birkhoff polytope.
	www.math.binghamton.edu /dennis/Papers/birkhoff.html (180 words)

	ehrhart - Around2.co.uk (Site not responding. Last check: 2007-10-31)
		Ehrhart Polynomial -- from MathWorld Ehrhart Polynomial -- from MathWorld Let \Delta denote an integral convex polytope of dimension n in a lattice M, and let l_\Delta(k) denote the number of lattice points in \Delta dilated by a factor of the...
		Ehrhart showed in 1962 that L is a...
		Earl Ehrhart is the Chairman of the Rules Committee of the Georgia House of Representatives...
	www.around2.co.uk /directory/e/ehrhart/readme.htm (1278 words)

	[No title]
		Here we show that the Ehrhart polynomial of a lattice $n$-simplex has a simple analytical interpretation from the perspective of Fourier Analysis on the $n$-torus.
		Algebraic geometers have shown that the Hilbert polynomials of toric varieties associated to convex lattice polytopes precisely describe the number of lattice points inside their dilates \cite{3}.
		\end{equation} Ehrhart* \cite{4} inaugurated the systematic study of general properties of this function by proving that it is always a polynomial in $t \in \mathbb{N}$, and that in fact \begin{equation}\latticeP = \text{Vol} (\Pe) t^{n} + \frac{1}{2} \text{Vol} (\partial \Pe) t^{n-1} + \cdots + \chi (\Pe) \end{equation} for closed polytopes $\Pe $.
	www.math.psu.edu /era-mirror/1996-01-001/1996-01-001.tex.html (686 words)

	[No title]
		Application to chromatic polynomials (SKIP, IF TIME IS SHORT) Given a graph G with m vertices, let chi(n) be the number of ways to assign colors to the vertices of G so that no vertices of G that share an edge are assigned the same color, where the set of allowed colors has size n.
		Claim: chi(n) is a polynomial in n, and chi(-1) is (-1)^m times the number of acyclic orientations of G. Example: Let G be a cycle of length 4.
		Note that there might be exponentially many such pieces, as a function of the number of edges of G, but G is fixed: the number of pieces doesn’t vary as n varies.
	www.math.harvard.edu /~propp/192/12-06.doc (536 words)

	Julian Pfeifle
		The Ehrhart polynomial of a convex lattice polytope counts integer points in integral dilates of the polytope.
		We prove that for fixed d, there exists a bounded region of C containing all roots of Ehrhart polynomials of d-polytopes, and that all real roots of these polynomials lie in [-d, \lfloor d/2 \rfloor)$.
		We finish with an experimental investigation of the Ehrhart polynomials of cyclic polytopes and 0/1-polytopes.
	www-ma2.upc.edu /~julian (932 words)

	Ehrhart polynomial Details, Meaning Ehrhart polynomial Article and Explanation Guide
		In mathematics, integral polytopes have associated Ehrhart polynomials which encode some geometrical information about them.
		Formulas for the other coefficients are much harder to get; Todd classes of toric varieties, the Riemann-Roch theorem as well as Fourier analysis have been used for this purpose.
		Ricardo Diaz, Sinai Robins: The Ehrhart polynomial of a lattice n-simplex, Electronic Research Announcements of the American Mathematical Society 2 (1996), pages 1-6, online version.
	www.e-paranoids.com /e/eh/ehrhart_polynomial.html (316 words)

	Interface between MuPAD and Polylib
		We use it here to compute the Ehrhart polynomial of a polyhedron defined under MuPAD.
		The so-called "Ehrhart polynomial" gives a closed form to the number of integer points in a polyhedron defined by a set of inequalities; the polyhedron is allowed to be defined in terms of symbolic parameters.
		Here _periodic takes 2 parameters: a list l and a parameter p; the result is the k-th element of the list (counting up from 0), where k=p (modulo length of list).
	www-rocq.inria.fr /who/Francois.Thomasset/mupad-ehrhart/README.html (551 words)

	Ehrhart polynomial: Facts and details from Encyclopedia Topic (Site not responding. Last check: 2007-10-31)
		Ehrhart showed in 1967 that L is a rational polynomial (A mathematical expression that is the sum of a number of terms)
		The Ehrhart polynomial of the interior (The region that is inside of something)
		Minimal polynomial (The minimal polynomial of an n-by-n matrix a over a field (mathematics)field f is the monic...)
	www.absoluteastronomy.com /ref/ehrhart_polynomial (1164 words)

	[No title] (Site not responding. Last check: 2007-10-31)
		It is well known that this parametric count can be represented by a set of Ehrhart polynomials.
		Its worst-case computation time for a single Ehrhart polynomial is exponential in the input size, even for fixed dimensions.
		The worst-case size of such an Ehrhart polynomial (measured in bits needed to represent the polynomial) is also exponential in the input size.
	escher.elis.ugent.be /p/abstract.php?klnr=P104.090 (182 words)

	David A. Cox
		The topics for Fall 2004 are the algebra and geometry of polynomial equations.
		These are notes for three lectures covering various ways to define of toric varieties (fans, homogeneous coordinates, and toric ideals), results related to polytopes (the Dehn-Sommerville equations, the Ehrhart polynomial, and the BKK bound), and an introduction to how toric varieties are used in Mirror Symmetry (the Batyrev mirror construction).
		The minicourse was given at the University of Buenos Aires in the Summer of 2001.
	www.amherst.edu /~dacox (867 words)

	Example 1, PRDG of QR (Site not responding. Last check: 2007-10-31)
		The evaluation of the polynomial is done by the parameter evaluation mechanism present in Ptolemy II.
		If we change the parameter values, the Ehrhart polynomials are being re-evaluated, given the new firing and communication values very quickly.
		However, to allows a designer to quickly show where the hot-spots are in a program, the edges and nodes are colored according to a color scheme running from blue to red in 15 different color shades.
	ptolemy.eecs.berkeley.edu /~kienhuis/compaan/web/panda_qr.htm (439 words)

	Citebase - The Ehrhart polynomial of the Birkhoff polytope (Site not responding. Last check: 2007-10-31)
		One reason to be interested in this counting function is that the leading term of the Ehrhart polynomial is--up to a trivial factor--the volume of the polytope.
		Citation coverage and analysis is incomplete and hit coverage and analysis is both incomplete and noisy.
		The coefficients are shown to be given by polynomials in λ, μ and ν on the cones of the chamber complex of a vector partition function.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0202267 (1166 words)

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