The rational homology groups of packing complexes are important in algebraic geometry since they control the syzygies of line bundles on projective embeddings of products of projective spaces (Segre–Veronese varieties). These complexes are a common generalization of the multidimensional chessboard complexes and of the matching complexes of complete uniform hypergraphs, whose study has been a topic of interest in combinatorial topology. We prove that the multivariate version of representation stability,...

We consider the representation theory for a class of log-canonical surface singularities in the sense of reflexive (or equivalently maximal Cohen-Macaulay) modules and in the sense of finite dimensional representations of the local fundamental group. A detailed classification and enumeration of the indecomposable reflexive modules is given, and we prove that any reflexive module admits an integrable connection and hence is induced from a finite dimensional representation of the local fundamental...

Consider a compact subset $K$ of real $n$-space defined by polynomial inequalities $g_1\ge 0,\cdots ,g_s\ge 0$. For a polynomial $f$ non-negative on $K$, natural sufficient conditions are given (in terms of first and second derivatives at the zeros of $f$ in $K$) for $f$ to have a presentation of the form $f=t_0+t_1{g}_1+\cdots +t_s{g}_s$, $t_i$ a sum of squares of polynomials. The conditions are much less restrictive than the conditions given by Scheiderer in [11, Cor. 2.6]. The proof uses Scheiderer’s main theorem in [11] as well as arguments from quadratic form theory...

This paper studies the representation of a non-negative polynomial f on a non-compact semi-algebraic set K modulo its KKT (Karush-Kuhn-Tucker) ideal. Under the assumption that f satisfies the boundary Hessian conditions (BHC) at each zero of f in K, we show that f can be represented as a sum of squares (SOS) of real polynomials modulo its KKT ideal if f ≥ 0 on K.

We show an explicit relation between the Chow form and the Grothendieck residue; and we clarify the role that the residue can play in the intersection theory besides its role in the division problem.

Let 𝓐(ℝ) and 𝓔(ℝ) denote respectively the ring of analytic and real entire functions in one variable. It is shown that if 𝔪 is a maximal ideal of 𝓐(ℝ), then 𝓐(ℝ)/𝔪 is isomorphic either to the reals or a real closed field that is an η₁-set, while if 𝔪 is a maximal ideal of 𝓔(ℝ), then 𝓔(ℝ)/𝔪 is isomorphic to one of the latter two fields or to the field of complex numbers. Moreover, we study the residue class rings of prime ideals of these rings and their Krull dimensions. Use is made of...

On considère l’espace de modules $M(c_1,c_2)$ des fibrés stables de rang $2$ sur ${\mathbb{P}}_k^3$, de classes de Chern $c_1,c_2$, $k$ étant un corps algébriquement clos de caractéristique quelconque. Si ($c_1=0,c_2\>0$) ou ($c_1=-1,c_2=2p\ge 6$), on sait ([7], [9]) que $M(c_1,c_2)$ a une composante irréductible dont le point générique $ℱ(c_1,c_2)$ a la cohomologie naturelle. Nous avons calculé ([16]) la résolution minimale de $ℱ(0,c_2)$. Dans cet article, nous voulons déterminer celle de $ℱ(-1,c_2)$ si ${\displaystyle c_2\>\frac{(v+2)(2v^2+3v-1)}{6v+7},}$ où $v$ est le plus petit entier tel que $h^0\left(ℱ\left(v\right)\right)\>0$. Par un procédé standard rappelé dans [16], on se ramène à des...