Related papers: The ${\rm Jacobian Conjecture}_{2n}$ implies the $…

We reveal a complex analogue to a result about polynomial solutions to the Dirichlet Problem on ellipsoids in $\mathbb{R}^n$ by showing that the Bergman projection on any ellipsoid in $\mathbb{C}^n$ is such that the projection of any…

The well-known Lvov-Kaplansky conjecture states that the image of a multilinear polynomial $f$ evaluated on $n\times n$ matrices is a vector space. A weaker version of this conjecture, known as the Mesyan conjecture, states that if $m=deg(…

In the paper {\em The inversion formulae for automorphisms of polynomial algebras and differential operators in prime characteristic}, J. Pure Appl. Algebra 212 (2008), no. 10, 2320-2337, see also arXiv:math/0604477, Vladimir Bavula states…

The said paper [2] entitled "Proof Of Two Dimensional Jacobian Conjecture" is with gaps.

Let $F:\Bbb C^n\to\Bbb C^n$ be a polynomial mapping with a non vanishing Jacobian. If the set $S_F$ of non-properness of $F$ is smooth, then $F$ is a surjective mapping. Moreover, the set $S_F$ can not be connected (this is the…

The Jacobian Conjecture would follow if it were known that real polynomial maps with a unipotent Jacobian matrix are injective. The conjecture that this is true even for $C^1$ maps is explored here. Some results known in the polynomial case…

In this paper we prove the Gromov--Milman conjecture (the Dvoretzky type theorem) for homogeneous polynomials on $\mathbb R^n$, and improve bounds on the number $n(d,k)$ in the analogous conjecture for odd degrees $d$ (this case is known as…

This article is part of an ongoing investigation of the two-dimensional Jacobian conjecture. In the first paper of this series, we proved the generalized Magnus' formula. In this paper, inspired by cluster algebras, we introduce a sequence…

Let K[x,y] be the algebra of two-variable polynomials over a field K. A polynomial p=p(x, y) is called a test polynomial (for automorphisms) if, whenever \phi(p)=p for a mapping \phi of K[x,y], this \phi must be an automorphism. Here we…

The Newton polytope related to a ``minimal" counterexample to the Jacobian conjecture is introduced and described. This description allows to obtain a sharper estimate for the geometric degree of the polynomial mapping given by a Jacobian…

We prove and conjecture results which show that Castelnuovo theory in projective space has a close analogue for abelian varieties. This is related to the geometric Schottky problem: our main result is that a principally polarized abelian…

In the large rank limit, for any nonexceptional affine algebra, the graded branching multiplicities known as one-dimensional sums, are conjectured to have a simple relationship with those of type A, which are known as generalized Kostka…

We obtain similar types of conclusions as that of Br\"{u}ck [1] for two differential polynomials which in turn radically improve and generalize several existing results. Moreover, a number of examples have been exhibited to justify the…

In an earlier work [K. Castillo et al., J. Math. Anal. Appl., 514 (2022) 126358], we give positive answer to the first, and apparently more easy, part of a conjecture of M. Ismail concerning the characterization of the continuous $q$-Jacobi…

We obtain a structure theorem for the nonproperness set $S_f$ of a nonsingular polynomial mapping $f:\mathbb{C}^n \to \mathbb{C}^n$. Jelonek's results on $S_f$ and our result show that if $f$ is a counterexample to the Jacobian conjecture,…

Let $f=(f_1, f_2)$ be a regular sequence of affine curves in $\bC^2$. Under some reduction conditions achieved by composing with some polynomial automorphisms of $\bC^2$, we show that the intersection number of curves $(f_i)$ in $\bC^2$…

We prove a generalization of a conjecture of Dokos, Dwyer, Johnson, Sagan, and Selsor giving a recursion for the inversion polynomial of 321-avoiding permutations. We also answer a question they posed about finding a recursive formulas for…

In this paper, we will give suitable conditions on differential polynomials $Q(f)$ such that they take every finite non-zero value infinitely often, where $f$ is a meromorphic function in complex plane. These results are related to Problem…

In this paper we show that the image of any locally finite $k$-derivation of the polynomial algebra $k[x, y]$ in two variables over a field $k$ of characteristic zero is a Mathieu subspace. We also show that the two-dimensional Jacobian…

In 2018 Kashaev introduced a diagrammatic link invariant conjectured to be twice the Levine-Tristram signature. If true, the conjecture would provide a simple way of computing the Levine-Tristram signature of a link by taking the signature…