“Ehrhart polynomial”的意思、由来-开放百科全书

In mathematics, an integral polytope has an associated Ehrhart polynomial that encodes the relationship between the volume of a polytope and the number of integer points the polytope contains. The theory of Ehrhart polynomials can be seen as a higher-dimensional generalization of Pick's theorem in the Euclidean plane.

These polynomials are named after Eugène Ehrhart who studied them in the 1960s.

Definition

Informally, if {{math|P}} is a polytope, and {{math|tP}} is the polytope formed by expanding {{math|P}} by a factor of {{math|t}} in each dimension, then {{math|L(P, t)}} is the number of integer lattice points in {{math|tP}}.

More formally, consider a lattice

in Euclidean space {{math|ℝⁿ}} and a {{math|d}}-dimensional polytope {{math|P}} in {{math|ℝⁿ}} with the property that all vertices of the polytope are points of the lattice. (A common example is

and a polytope for which all vertices have integer coordinates.) For any positive integer {{math|t}}, let {{math|tP}} be the {{math|t}}-fold dilation of {{math|P}} (the polytope formed by multiplying each vertex coordinate, in a basis for the lattice, by a factor of {{math|t}}), and let

be the number of lattice points contained in the polytope {{math|tP}}. Ehrhart showed in 1962 that {{math|L}} is a rational polynomial of degree {{math|d}} in {{math|t}}, i.e. there exist rational numbers

such that:

The Ehrhart polynomial of the interior of a closed convex polytope {{math|P}} can be computed as:

where {{math|d}} is the dimension of {{math|P}}. This result is known as Ehrhart–Macdonald reciprocity.^[1]

Examples of Ehrhart polynomials

Let {{math|P}} be a {{math|d}}-dimensional unit hypercube whose vertices are the integer lattice points all of whose coordinates are 0 or 1. In terms of inequalities,

Then the {{math|t}}-fold dilation of {{math|P}} is a cube with side length {{math|t}}, containing {{math|(t + 1)^d}} integer points. That is, the Ehrhart polynomial of the hypercube is {{math|L(P,t) {{=}} (t + 1)^d}}.^[2]^[3] Additionally, if we evaluate {{math|L(P, t)}} at negative integers, then

Many other figurate numbers can be expressed as Ehrhart polynomials. For instance, the square pyramidal numbers are given by the Ehrhart polynomials of a square pyramid with an integer unit square as its base and with height one; the Ehrhart polynomial in this case is {{math|{{sfrac|1|6}}(t + 1)(t + 2)(2t + 3)}}.^[4]

Ehrhart quasi-polynomials

where

and

. (Equivalently, {{math|P}} is the convex hull of finitely many points in {{math|ℚ^d}}.) Then define

In this case, {{math|L(P, t)}} is a quasi-polynomial in {{math|t}}. Just as with integral polytopes, Ehrhart–Macdonald reciprocity holds, that is,

Examples of Ehrhart quasi-polynomials

Let {{math|P}} be a polygon with vertices (0,0), (0,2), (1,1) and ({{sfrac|3|2}}, 0). The number of integer points in {{math|tP}} will be counted by the quasi-polynomial ^[5]

Interpretation of coefficients

If {{math|P}} is closed (i.e. the boundary faces belong to {{math|P}}), some of the coefficients of {{math|L(P, t)}} have an easy interpretation:

The Betke–Kneser theorem

Ulrich Betke and Martin Kneser^[6] established the following characterization of the Ehrhart coefficients. A functional

defined on integral polytopes is an

and translation invariant valuation if and only if there are

such that

Ehrhart series

We can define a generating function for the Ehrhart polynomial of an integral {{math|d}}-dimensional polytope {{math|P}} as

This series can be expressed as a rational function. Specifically, Ehrhart proved (1962){{cn|date=February 2017}} that there exist complex numbers {{math|h{{su|b=i|p=*}}}}, {{math|0 ≤ j ≤ d}}, such that the Ehrhart series of {{math|P}} is

Additionally, Stanley's non-negativity theorem states that under the given hypotheses, {{math|h{{su|b=i|p=*}}}} will be non-negative integers, for {{math|0 ≤ j ≤ d}}.

Another result by Stanley{{cn|date=February 2017}} shows that if {{math|P}} is a lattice polytope contained in {{math|Q}}, then {{math|h{{su|b=i|p=*}}(P) ≤ h{{su|b=i|p=*}}(Q)}} for all {{math|i}}.

The {{math|h^*}}-vector is in general not unimodal, but it is whenever it is symmetric, and the polytope has a

Ehrhart series for rational polytopes

As in the case of polytopes with integer vertices, one defines the Ehrhart series for a rational polytope. For a rational polytope {{math|P}}, where {{math|d}} is the smallest integer such that {{math|dP}} is an integer polytope ({{math|d}} is called the denominator of {{math|P}}), then one has

Toric variety

The case

and

of these statements yields Pick's theorem. 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.

If {{math|X}} is the toric variety corresponding to the normal fan of {{math|P}}, then {{math|P}} defines an ample line bundle on {{math|X}}, and the Ehrhart polynomial of {{math|P}} coincides with the Hilbert polynomial of this line bundle.

Ehrhart polynomials can be studied for their own sake. For instance, one could ask questions related to the roots of an Ehrhart polynomial.^[10] Furthermore, some authors have pursued the question of how these polynomials could be classified.^[11]

Generalizations

It is possible to study the number of integer points in a polytope {{math|P}} if we dilate some facets of {{math|P}} but not others. In other words, one would like to know the number of integer points in semi-dilated polytopes. It turns out that such a counting function will be what is called a multivariate quasi-polynomial. An Ehrhart-type reciprocity theorem will also hold for such a counting function.^[12]

Counting the number of integer points in semi-dilations of polytopes has applications ^[13] in enumerating the number of different dissections of regular polygons and the number of non-isomorphic unrestricted codes, a particular kind of code in the field of coding theory.

See also

Notes

1. ^{{cite journal|last=Macdonald|first=Ian G.|authorlink=Ian G. Macdonald| title=Polynomials associated with finite cell-complexes|journal=Journal of the London Mathematical Society|year=1971|volume=2|issue=1|pages=181–192|doi=10.1112/jlms/s2-4.1.181 |url=http://jlms.oxfordjournals.org/content/s2-4/1/181.full.pdf}}
2. ^{{harvtxt|De Loera|Rambau|Santos|2010}}
3. ^{{harvtxt|Mathar|2010}}
4. ^{{harvtxt|Beck|De Loera|Develin|Pfeifle|2005}}.
5. ^{{cite book|title=Computing the Continuous Discretely|year=2007|publisher=Springer|location=New York|pages=46–47|last=Beck|first=Matthias|last2=Robins|first2=Sinai|MR=2271992}}
6. ^{{citation|last1=Betke|first1= Ulrich|last2= Kneser|first2=Martin|authorlink2=Martin Kneser| year=1985|title=Zerlegungen und Bewertungen von Gitterpolytopen|journal= Journal für die reine und angewandte Mathematik |volume=358|pages= 202–208|mr=0797683}}
7. ^{{cite journal|last1=Athanasiadis|first1=Christos A.|title=h*-Vectors, Eulerian Polynomials and Stable Polytopes of Graphs|journal=Electronic Journal of Combinatorics|year=2004|volume=11|issue=2|url=http://www.combinatorics.org/ojs/index.php/eljc/article/view/v11i2r6}}
8. ^{{cite journal|last=Stanley|first=Richard P.|authorlink=Richard P. Stanley|title=Decompositions of rational convex polytopes|journal=Annals of Discrete Mathematics|date=1980|volume=6|pages=333–342|doi=10.1016/s0167-5060(08)70717-9}}
9. ^{{cite journal|last1=Beck|first1=Matthias|last2=Sottile|first2=Frank|title=Irrational proofs for three theorems of Stanley|journal=European Journal of Combinatorics|date=January 2007|volume=28|issue=1|pages=403–409|doi=10.1016/j.ejc.2005.06.003|arxiv=math/0501359}}
10. ^{{cite journal|last=Braun|first=Benjamin|last2=Develin|first2=Mike|authorlink2=Mike Develin| title=Ehrhart Polynomial Roots and Stanley's Non-Negativity Theorem|publisher=American Mathematical Society |year=2008|volume=452|series=Contemporary Mathematics|pages=67–78|doi=10.1090/conm/452/08773|arxiv=math/0610399}}
11. ^{{cite journal|last=Higashitani|first=Akihiro|title=Classification of Ehrhart Polynomials of Integral Simplices|journal=DMTCS Proceedings|year=2012|pages=587–594|url=http://www.math.nagoya-u.ac.jp/fpsac12/download/contributed/dmAR0152.pdf}}
12. ^{{cite journal|last=Beck|first=Matthias|title=Multidimensional Ehrhart reciprocity|journal=Journal of Combinatorial Theory|date=January 2002|volume=97|series=Series A|issue=1|pages=187–194|url=http://www.sciencedirect.com/science/article/pii/S0097316501932200|doi=10.1006/jcta.2001.3220}}
13. ^{{cite journal|last=Lisonek|first=Petr|title=Combinatorial Families Enumerated by Quasi-polynomials|journal=Journal of Combinatorial Theory|year=2007|volume=114|series=Series A|issue=4|pages=619–630|url=http://www.sciencedirect.com/science/article/pii/S0097316506001427|doi=10.1016/j.jcta.2006.06.013}}