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The third Russian-Chinese conference on complex analysis, algebra, algebraic geometry and mathematical physics
May 12, 2016 13:10–13:40, Moscow, Steklov Mathematical Institute of RAS, Gubkina, 8

McMullen's formula and a multidimensional analog of the Weierstrass $\zeta$-function

A. V. Shchuplev

Institute of Mathematics and Computer Science, Siberian Federal University, Krasnoyarsk
 Video records: Flash Video 173.9 Mb Flash Video 1,036.5 Mb MP4 173.9 Mb

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Abstract: In 1899 G. Pick found a simple formula relating the area $\mathrm{Area}(P)$ of a plane polygon $P$ with vertices in integer points with the number $I$ of its interior points and on its boundary $B$:
$$\mathrm{Area}(P)=I+\frac{B}{2}-1.$$
However, this formula can not be simply extended even to three-dimensional case as Reeve's example demonstrates.Instead of one simple formula there exist several formulas, obtained by combinatorial or number-theoretical methods, or by methods of algebraic geometry. One such formula due to P.McMullen says that for an integer polyhedron with centrally-symmetric facets its volume is equal to the sum of all solid angles at each its integer point of the polyhedron.
A multidimensional analog of the Weierstrass $\zeta$-function
$$\zeta(z)=\eta(z)+\sum_{\gamma\in{\mathbb Z^{2n}\smallsetminus \{0\}}}(\eta(z-\gamma)+\eta(\gamma)+\sum_{i=1}^{n}(\frac{\partial\eta}{\partial z_{i}}(\gamma)z_{i}+\frac{\partial\eta}{\partial\bar{z}_{i}}(\gamma)\bar{z}_{i})),$$
where $\eta(z)$ is the Bochner-Martinelli differential form allows to prove this statement analytically.

Language: English

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