Related papers: Towards Common Zeros of Iterated Morphisms

Jedrzejewicz showed that a polynomial map over a field of characteristic zero is invertible, if and only if the corresponding endomorphism maps irreducible polynomials to irreducible polynomials. Furthermore, he showed that a polynomial map…

We show that the problem of classifying, up to isomorphism, the collection of zero-entropy mixing automorphisms of a standard non-atomic probability space, is intractible. More precisely, the collection of isomorphic pairs of automorphisms…

We study the zero-sharing behavior among irreducible characters of a finite group. For symmetric groups $S_n$, it is proved that, with one exception, any two irreducible characters have at least one common zero. To further explore this…

We generalize the classical Tanaka result on the finiteness of symmetry algebra for non-degenerate pseudo-product structures to the case when the completely-integrable distributions defining the pseudo-product structure are no longer…

Pairings are particular bilinear maps, and as any bilinear maps they factor through the tensor product as group homomorphisms. Besides, nothing seems to prevent us to construct pairings on other abelian groups than elliptic curves or more…

The Zariski theorem says that for every hypersurface in a complex projective (resp. affine) space of dimension at least 3 and for every generic plane in the projective (resp. affine) space the natural embedding generates an isomorphism of…

In this paper we investigate iteration of maps on lattices and the corresponding polynomial-like iterative equation. Since a lattice need not have a metric space structure, neither the Schauder fixed point theorem nor the Banach fixed point…

About two decades ago, Tsfasman and Boguslavsky conjectured a formula for the maximum number of common zeros that $r$ linearly independent homogeneous polynomials of degree $d$ in $m+1$ variables with coefficients in a finite field with $q$…

In a previous work, the third named author found a combinatorics of line arrangements whose realizations live in the cyclotomic group of the fifth roots of unity and such that their non-complex-conjugate embedding are not topologically…

We study the Zariski topology of the ind-groups of polynomial and free associative algebras $\Aut(K[x_1,...,x_n])$ (which is equivalent to the automorphism group of the affine space $\Aut(K^n))$) and $\Aut(K< x_1,..., x_n>$ via…

In this paper, complement-equivalent arithmetic Zariski pairs will be exhibited answering in the negative a question by Eyral-Oka on these curves and their groups. A complement-equivalent arithmetic Zariski pair is a pair of complex…

A "tensor space" is a vector space equipped with a finite collection of multi-linear forms. In previous work, we showed that (for each signature) there exists a universal homogeneous tensor space, which is unique up to isomorphism. Here we…

We show that all non-trivial continuous endomorphisms of the circle group are topologically mixing. We also show that there exists a large infinite class of continuous endomorphisms of any n-dimensional torus group which are topologically…

We address the question of finding algebraic properties that are respectively equivalent, for a morphism between algebraic varieties over an algebraically closed field of characteristic zero, to be an homeomorphism for the Zariski topology…

We investigate sets of the common zeros of non-constant semi-invariants for regular modules over canonical algebras. In particular, we show that if the considered algebra is tame then for big enough vectors these sets are complete…

On Zariski Main Theorem in Algebraic Geometry and Analytic Geometry. We fill a surprising gap of Complex Analytic Geometry by proving the analogue of Zariski Main Theorem in this geometry, i.e. proving that an holomorphic map from an…

Let G denote either a special orthogonal group or a symplectic group defined over the complex numbers. We prove the following saturation result for G: given dominant weights \lambda^1, ..., \lambda^r such that the tensor product…

Motivated by the study of unlikely intersection in the moduli space of rational maps, we initiate our investigation on algebraic dynamics for families of regular polynomial skew products in this article. Our goals are threefold. (1) We…

We introduce the first provably efficient algorithm to check if a finitely generated subgroup of an almost simple semi-simple group over the rationals is Zariski-dense. We reduce this question to one of computing Galois groups, and to this…

Let $f \colon X \to X$ be a surjective endomorphism of a normal projective surface. When $\operatorname{deg} f \geq 2$, applying an (iteration of) $f$-equivariant minimal model program (EMMP), we determine the geometric structure of $X$.…