Related papers: On a theorem of Brion

The finite Pfaff lattice is given by commuting Lax pairs involving a finite matrix L (zero above the first subdiagonal) and a projection onto Sp(N). The lattice admits solutions such that the entries of the matrix L are rational in the time…

We give a new proof of Fatou's theorem: {\em if an algebraic function has a power series expansion with bounded integer coefficients, then it must be a rational function.} This result is applied to show that for any non--trivial completely…

We introduce the concept of a bounded below set in a lattice. This can be used to give a generalization of Rota's broken circuit theorem to any finite lattice. We then show how this result can be used to compute and combinatorially explain…

In [1] M. Baker and S. Norine developed a theory of divisors and linear systems on graphs, and proved a Riemann-Roch Theorem for these objects (conceived as integer-valued functions on the vertices). In [2] and [3] the authors generalized…

We present a simple proof of the fact that the base (and independence) polytope of a rank $n$ regular matroid over $m$ elements has an extension complexity $O(mn)$.

Suppose p is a symmetric matrix whose entries are polynomials in freely noncommutating variables and p(0) is positive definite. Let D(p) denote the component of zero of the set of those g-tuples X of symmetric matrices (of the same size)…

We prove a result that enables us to calculate the rational homotopy of a wide class of spaces by the theory of minimal models.

If $R$ is a rational map, the Main Result is a uniformization Theorem for the space of decompositions of the iterates of $R$. Secondly, we show that Fatou conjecture holds for decomposable rational maps.

We study the algebraic properties of binary relations whose underlying digraph is smooth, that is has no source or sink. Such objects have been studied as surjective hyper-operations (shops) on the corresponding vertex set, and as binary…

Faces play a central role in the combinatorial and computational aspects of polyhedra. In this paper, we present the first formalization of faces of polyhedra in the proof assistant Coq. This builds on the formalization of a library…

This is an anthology of series involving rational, factorial, and power functions expressed in terms of special functions. New finite expansions involving quotient functions expressed in terms of the Hurwitz-Lerch zeta function are given.…

Let $f(z) = \sum_{k=0}^\infty d_k z^k$, $d_k\in\mathbb{C}\backslash\{ 0 \}$, $d_0=1$, be a power series with a non-zero radius of convergence $\rho$: $0 <\rho \leq +\infty$. Denote by $f_n(z)$ the n-th partial sum of $f$, and $R_{2n}(z) =…

In this paper, a new proof of the Positive Mass Theorem is established through a newly discovered monotonicity formula, holding along the level sets of the Green's function of an asymptotically flat $3$-manifold. In the same context and for…

Recently, Chapoton found a $q$-analog of Ehrhart polynomials, which are polynomials in $x$ whose coefficients are rational functions in $q$. Chapoton conjectured the shape of the Newton polygon of the numerator of the $q$-Ehrhart polynomial…

We relate a previous result of ours on families of Diophantine equations having only trivial solutions with a result on the approximation of an algebraic number by products of rational numbers and units. We compare this approximation with a…

Given any polytope $P$ and any generic linear functional ${\bf c} $, one obtains a directed graph $G(P,{\bf c})$ from the 1-skeleton of $P$ by orienting each edge $e(u,v)$ from $u$ to $v$ for ${\bf c} (u) < {\bf c} ( v)$. For $P$ a simple…

We present a compact formula computing rational trigonometric sums. E. Verlinde's expression for the dimension of conformal blocks in WZW theory is an example of such a sum. As an application, we show that a formula of Bismut and Labourie…

We introduce the property of convex normality of rational polytopes and give a dimensionally uniform lower bound for the edge lattice lengths, guaranteeing the property. As an application, we show that if every edge of a lattice d-polytope…

We introduce and study the algebraic, analytic and lattice properties of regular homogeneous polynomials and holomorphic functions on complex Banach lattices. We show that the theory of power series with regular terms is closer to the…

Let X be a projective cubic hypersurface of dimension 11 or more, which is defined over the rationals. In this paper it is shown that X contains rational points provided that the cubic form defining X can be written as the sum of two forms…