"Quasipolynomial"의 두 판 사이의 차이
imported>Pythagoras0 |
Pythagoras0 (토론 | 기여) |
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| 1번째 줄: | 1번째 줄: | ||
==example== | ==example== | ||
| − | * assume | + | * assume <math>a_n = \left((-1)^n+1\right)+\left((-1)^n+3\right) n</math> |
* then | * then | ||
| − | + | :<math> | |
\sum_{n=0}^{\infty}a_nt^n = \frac{2 \left(t^3+3 t^2+t+1\right)}{(1-t)^2 (t+1)^2} | \sum_{n=0}^{\infty}a_nt^n = \frac{2 \left(t^3+3 t^2+t+1\right)}{(1-t)^2 (t+1)^2} | ||
| − | + | </math> | |
==some results== | ==some results== | ||
;thm (Ehrhart's theorem for rational polytopes) | ;thm (Ehrhart's theorem for rational polytopes) | ||
| − | If | + | If <math>P</math> is a rational convex <math>d</math>-polytope, then <math>L_{P}(t)</math> is a quasipolynomial in <math>t</math> of degree <math>d</math>. Its period divides the least common multiple of the denominator of the coordinates of the vertices of <math>P</math>. |
;lemma (Beck-Robins ex. 3.19) | ;lemma (Beck-Robins ex. 3.19) | ||
| − | If | + | If <math>\sum_{t \ge 0} f(t)z^t = \frac{g(z)}{h(z)}</math>, then <math>f</math> is a quasipolynomial of degree <math>d</math> with period <math>p</math> if and only if <math>g</math> and <math>h</math> are polynomials such that <math>\deg(g)<\deg(h)</math>, all roots of <math>h</math> are <math>p</math>-th roots of unity of multiplicity at most <math>d+1</math>, and there is a root of multiplicity equal to <math>d+1</math> (all of this assuming that <math>g/h</math> has been reduced to lowest terms. |
;thm (Beck-Robins ex. 3.25) | ;thm (Beck-Robins ex. 3.25) | ||
| − | Suppose | + | Suppose <math>P</math> is a rational <math>d</math>-polytope with denominator <math>p</math>. Then |
| − | + | :<math> | |
\operatorname{Ehr}_{P}(z) = \frac{f(z)}{(1-z^p)^{d+1}} | \operatorname{Ehr}_{P}(z) = \frac{f(z)}{(1-z^p)^{d+1}} | ||
| − | + | </math> | |
| − | where | + | where <math>f</math> is a polynomial with nonnegative integral coefficients. |
;prop (?) | ;prop (?) | ||
| − | Let | + | Let <math>f</math> be a quasipolynomial of degree <math>d</math>. If <math>f(n+1)\geq f(n)</math> for all <math>n\in \mathbb{N}</math>, then the top degree coefficient of <math>f</math> must be constant. |
[[분류:migrate]] | [[분류:migrate]] | ||
2020년 11월 16일 (월) 04:25 기준 최신판
example
- assume \(a_n = \left((-1)^n+1\right)+\left((-1)^n+3\right) n\)
- then
\[ \sum_{n=0}^{\infty}a_nt^n = \frac{2 \left(t^3+3 t^2+t+1\right)}{(1-t)^2 (t+1)^2} \]
some results
- thm (Ehrhart's theorem for rational polytopes)
If \(P\) is a rational convex \(d\)-polytope, then \(L_{P}(t)\) is a quasipolynomial in \(t\) of degree \(d\). Its period divides the least common multiple of the denominator of the coordinates of the vertices of \(P\).
- lemma (Beck-Robins ex. 3.19)
If \(\sum_{t \ge 0} f(t)z^t = \frac{g(z)}{h(z)}\), then \(f\) is a quasipolynomial of degree \(d\) with period \(p\) if and only if \(g\) and \(h\) are polynomials such that \(\deg(g)<\deg(h)\), all roots of \(h\) are \(p\)-th roots of unity of multiplicity at most \(d+1\), and there is a root of multiplicity equal to \(d+1\) (all of this assuming that \(g/h\) has been reduced to lowest terms.
- thm (Beck-Robins ex. 3.25)
Suppose \(P\) is a rational \(d\)-polytope with denominator \(p\). Then \[ \operatorname{Ehr}_{P}(z) = \frac{f(z)}{(1-z^p)^{d+1}} \] where \(f\) is a polynomial with nonnegative integral coefficients.
- prop (?)
Let \(f\) be a quasipolynomial of degree \(d\). If \(f(n+1)\geq f(n)\) for all \(n\in \mathbb{N}\), then the top degree coefficient of \(f\) must be constant.