Related papers: Complete polynomial vector fields on $\C^2$, Part …
In the present paper, we first give a characterization for Bongartz completion in $\tau$-tilting theory via $c$-vectors. Motivated by this characterization, we give the definition of Bongartz completion in cluster algebras using…
In this paper, we first give two fundamental principles under a technique to characterize conformal vector fields of $(\alpha,\beta)$ spaces to be homothetic and determine the local structure of those homothetic fields. Then we use the…
We give an algorithm for deciding whether a planar polynomial differential system has a first integral which factorizes as a product of defining polynomials of curves with only one place at infinity. In the affirmative case, our algorithm…
In this paper, we consider the following two algebraic hypersurfaces $$S^1\times S^2=\{(x_1,x_2,x_3,x_4)\in \mathbb{R}^4:(x_1^2+x_2^2-a^2)^2 + x_3^2 + x_4^2 -1=0;~ a>1\}$$ and $$S^2\times S^1=\{(x_1,x_2,x_3,x_4)\in…
We show factorization of polynomials in one variable over the tropical semiring is in general NP-complete, either if all coefficients are finite, or if all are either 0 or infinity (Boolean case). We give algorithms for the factorization…
We construct the vector fields associated to the space-time invariances of relativistic particle theory in flat Euclidean space-time. We show that the vector fields associated to the massive theory give rise to a differential operator…
Let $k$ be a perfect field and let $C_0:f=0$ be a smooth curve in the torus $\mathbb{G}_{m,k}^2$. Let $\mathbb{T}_\Delta$ be the toric variety associated to the Newton polygon of $f$. Extending the toric resolution of $C_0$ on…
We study the growth of the central polynomials for the algebras $G$ and $M_k(F)$, the infinite dimensional Grassmann algebra and the $k\times k$ matrices over a field $F$ of characteristic zero. In particular it follows that $M_k(F)$…
We provide a complete classification of the singularities of cluster algebras of finite cluster type. This extends our previous work about the case of trivial coefficients. Additionally, we classify the singularities of cluster algebras for…
We classify, up to a natural equivalence relation, vector fields of the plane which belong to the kernel of a 1--form. This form can be closed, in which case the vector fields are integrable, or not, in which case the differential of the…
We show how to determine whether two given real polynomial functions of a single variable are Lipschitz equivalent by comparing the values and also the multiplicities of the given polynomial functions at their critical points. Then we show…
We consider some combinatorial problems on matrix polynomials over finite fields. Using results from control theory we give a proof of a result of Helmke, Jordan and Lieb on the number of linear unimodular matrix polynomials over a finite…
We study ordinary differential equations in the complex domain given by meromorphic vector fields on K\"ahler compact complex surfaces. We prove that if such an equation has a maximal single valued solution with Zariski-dense image (in…
In the paper we provide some polynomial identities for finite-dimensional algebras. A list of well known single polynomial identities is exposed and the classification of all $2$-dimensional algebras with respect to these identities is…
The paper proposes a vector generalization of the basic concepts of the theory of complex variable: the concept of modulus and argument of complex number. The author introduces some generalizations of the notion of holomorphic functions and…
Let G be a connected reductive algebraic group defined over an algebraically closed field of positive characteristic. We study a generalization of the notion of G-complete reducibility in the context of Steinberg endomorphisms of G. Our…
The set of all subspaces of a given dimension in a finite classical polar space has a structure of a symmetric association scheme. If the dimension is zero, this is the scheme of the collinearity graph of the space; If the dimension is…
We solve the problem of characterizing the existence of a polynomial matrix of fixed degree when its eigenstructure (or part of it) and some of its rows (columns) are prescribed. More specifically, we present a solution to the row (column)…
We study wildness of automorphisms of a polynomial ring in three variables in detail using the Shestakov-Umirbaev theory and its generalization.
Consider an analytical function $f:V\subset\mathbb R^2\rightarrow\mathbb R$ having $0$ as its regular value, a switching manifold $\Sigma=f^{-1}(0)$ and a piecewise analytical vector field $X=(X^+,X^-)$, i.e. $X^\pm$ are analytical vector…