This short note is a continuation of and and its purpose is to show that every simple zeropotent paramedial groupoid containing at least three elements is strongly balanced in the sense of .

In the famous paper [FZ2] Fomin and Zelevinsky obtained Cartan-Killing type classification of all cluster algebras of finite type, i.e. cluster algebras having only finitely many distinct cluster variables. A wider class of cluster algebras is formed by cluster algebras of finite mutation type which have finitely many exchange matrices (but are allowed to have infinitely many cluster variables). In this paper we classify all cluster algebras of finite mutation type with skew-symmetric exchange matrices....

Soient $k$ un corps commutatif et $I=(p_1,\cdots ,p_m)k_n\left[X\right]$ un idéal de l’anneau des polynômes $k[X_1,\cdots ,X_n]$ (éventuellement $I=k_n\left[X\right]$). Nous prouvons une conjecture de C. Berenstein - A. Yger qui affirme que pour tout polynôme $p$, élément de la clôture intégrale $\overline{I}$ de l’idéal $I$, on a une représentation$$p^m=\sum _{1\le i\le m}{p}_i{q}_i,\text{avec}maxdeg\left(q_i{p}_i\right)\le mdegp+m{d}_1\cdots d_m,$$où $d_i=deg{p}_i,1\le i\le m$.

On démontre que toute solution formelle $\bar{y}\left(x\right)$ d’un système d’équations analytiques réelles (resp. polynomiales réelles) $f(x,y)=0$, se relève en une solution ${\mathbf{C}}^{\infty }$ homotope à une solution analytique (resp. à une solution de Nash) aussi proche que l’on veut de $\bar{y}\left(x\right)$ pour la topologie de Krull. On utilise ce théorème pour démontrer l’algébricité (ou l’analyticité) de certains idéaux de $\mathbf{R}\left\{x\right\}$ (ou $\mathbf{R}\left[\right[x\left]\right]$), et aussi pour construire des déformations analytiques de germes d’ensembles analytiques en germes d’ensembles de Nash.

We introduce a novel application of Gröbner bases to solve (non-homogeneous) systems of integer linear equations over integers. For this purpose, we present a new algorithm which ascertains whether a linear system of equations has an integer solution or not; in the affirmative case, the general integer solution of the system is determined.

We study the effect of changing the residue field, on the topological properties of local algebra homomorphisms of analytic algebras (quotients of convergent power series rings). Although injectivity is not preserved, openness and closedness in the Krull topology, simple topology, and inductive topology is preserved.

We are concerned with solving polynomial equations over rings. More precisely, given a commutative domain A with 1 and a polynomial equation antn + ...+ a0 = 0 with coefficients ai in A, our problem is to find its roots in A.We show that when A = B[x] is a polynomial ring, our problem can be reduced to solving a finite sequence of polynomial equations over B. As an application of this reduction, we obtain a finite algorithm for solving a polynomial equation over A when A is F[x1, ..., xN] or F(x1,...

We consider Stanley-Reisner rings $k[x_1,...,x_n]/I\left(\mathscr{H}\right)$ where $I\left(\mathscr{H}\right)$ is the edge ideal associated to some particular classes of hypergraphs. For instance, we consider hypergraphs that are natural generalizations of graphs that are lines and cycles, and for these we compute the Betti numbers. We also generalize some known results about chordal graphs and study a weak form of shellability.