Related papers: Rational Degree Algebraic Geometry
We define a class of quadratic differential algebras which are generated as differential graded algebras by the elements of an Euclidean space. Such a differential algebra is a differential calculus over the quadratic algebra of its…
Differential algebraic geometry seeks to extend the results of its algebraic counterpart to objects defined by differential equations. Many notions, such as that of a projective algebraic variety, have close differential analogues but their…
We study algebraic subvarieties of strata of differentials in genus zero satisfying algebraic relations among periods. The main results are Ax-Schanuel and Andr\'e-Oort-type theorems in genus zero. As a consequence, one obtains several…
Linear Geometry studies geometric properties which can be expressed via the notion of a line. All information about lines is encoded in a ternary relation called a line relation. A set endowed with a line relation is called a liner. So,…
This survey paper was primarily written as as the support for a course pesented at the JNCF2025: it aims to present some material that illustrates the kind of estimates one can obtain in effective algebraic geometry, for affine polynomial…
This is the first installment of an exposition of an ACL2 formalization of elementary linear algebra, focusing on aspects of the subject that apply to matrices over an arbitrary commutative ring with identity, in anticipation of a future…
PhD dissertation consists in three lines of investigation involving rational elliptic surfaces, namely 1) a study of conic bundles on these surfaces; 2) an investigation of the possible intersection numbers of two sections and 3) a theorem…
In this expositional paper, we discuss commutative algebra -- a study inspired by the properties of integers, rational numbers, and real numbers. In particular, we investigate rings and ideals, and their various properties. After, we…
The aim of the paper is to discuss the relations between the three kinds of objects named in the title. In a sense, this is a survey of such relations; however, some new directions are also considered. This relates, especially, to sections…
In this work, we construct the algebra of differential forms with the cube of exterior differential equal to zero on one-dimensional space. We prove that this algebra is a graded q-differential algebra where q is a cubic root of unity.…
A number of models of linear logic are based on or closely related to linear algebra, in the sense that morphisms are "matrices" over appropriate coefficient sets. Examples include models based on coherence spaces, finiteness spaces and…
We explain connections among several, a priori unrelated, areas of mathematics: combinatorics, algebraic statistics, topology and enumerative algebraic geometry. Our focus is on discrete invariants, strongly related to the theory of…
A theory of graded manifolds can be viewed as a generalization of differential geometry of smooth manifolds. It allows one to work with functions which locally depend not only on ordinary real variables, but also on $\mathbb{Z}$-graded…
The degree polynomial of a multigraph $G$ is given by $\sum _{v \in V(G)} x^{\mbox{deg}(v)}$. We investigate here properties of the roots of such polynomials. In addition to examining the roots for some families of graphs with few and many…
To any directed graph we associate an algebra with edges of the graph as generators and with relations defined by all pairs of directed paths with the same origin and terminus. Such algebras are related to factorizations of polynomials over…
Computation of polynomial relative invariants is a classical tool in algebra. Relative differential invariants are central for the equivalence problem of geometric structures. We address the fundamental problem of finite generation of their…
In this article we associate a combinatorial differential graded algebra to a cubic planar graph G. This algebra is defined combinatorially by counting binary sequences, which we introduce, and several explicit computations are provided. In…
This paper studies the space of degree $d>1$ invariant q-laminations, i.e., geodesic laminations invariant under the $d$-tupling map of the circle and associated with equivalence relations. Our main construction associates a q-lamination…
Let $A$ and $B$ be finite-dimensional simple algebras with arbitrary signature over an algebraically closed field. Suppose $A$ and $B$ are graded by a semigroup $S$ so that the graded identitical relations of $A$ are the same as those of…
We extend the loop algebra construction for algebras graded by abelian groups to study graded-simple algebras over the field of real numbers (or any real closed field). As an application, we classify up to isomorphism the graded-simple…