The Algebraic Sets in the General Mathematics and Teaching Them

Most of the pictures in general mathematics are algebraic sets. Indeed, even the first figures taught in class 1 of elementary school are already algebraic sets or part of algebraic sets, such as lines and segments. Therefore, knowing with certainty the properties of algebraic sets is very important for good teaching of high school mathematics, and it is essential to teach them better. To give suggestions and help teachers teach mathematics more effectively, in this report, we will present the Zariski topology, some of their most important properties and the methods to teach algebraic sets.

An Upper Bound for the Size of $s$-Distance Sets in Real Algebraic Sets

In a recent paper, Petrov and Pohoata developed a new algebraic method which combines the Croot-Lev-Pach Lemma from additive combinatorics and Sylvester’s Law of Inertia for real quadratic forms. As an application, they gave a simple proof of the Bannai-Bannai-Stanton bound on the size of $s$-distance sets (subsets $\mathcal{A}\subseteq \mathbb{R}^n$ which determine at most $s$ different distances). In this paper we extend their work and prove upper bounds for the size of $s$-distance sets in various real algebraic sets. This way we obtain a novel and short proof for the bound of Delsarte-Goethals-Seidel on spherical s-distance sets and a generalization of a bound by Bannai-Kawasaki-Nitamizu-Sato on $s$-distance sets on unions of spheres. In our arguments we use the method of Petrov and Pohoata together with some Gröbner basis techniques.

Characterization of the symmetry class of an elasticity tensor using polynomial covariants

We formulate effective necessary and sufficient conditions to identify the symmetry class of an elasticity tensor, a fourth-order tensor which is the cornerstone of the theory of elasticity and a toy model for linear constitutive laws in physics. The novelty is that these conditions are written using polynomial covariants. As a corollary, we deduce that the symmetry classes are affine algebraic sets, a result which seems to be new. Meanwhile, we have been lead to produce a minimal set of 70 generators for the covariant algebra of a fourth-order harmonic tensor and introduce an original generalized cross-product on totally symmetric tensors. Finally, using these tensorial covariants, we produce a new minimal set of 294 generators for the invariant algebra of the elasticity tensor.