Related papers: Surjectivity of maps induced on matrices by polyno…

For $n\geq 2$, we consider the map on $M_n(\mathbb K)$ given by evaluation of a polynomial $f(X_1, \ldots, X_m)$ over the field $\mathbb K$. In this article, we explore the image of the diagonal map given by $f=\delta_1 X_1^{k_1} + \delta_2…

Let $R$ be an affine domain of characteristic zero with finite quotients. We prove that a polynomial map over $R$ is surjective if and only if it is surjective over $\hat{R_{\mathfrak{m}}}$, the completion of $R$ with respect to…

In this note we obtain the surjectivity of smooth maps into Euclidean spaces under mild conditions. As application we give a new proof of the Fundamental Theorem of Algebra. We also observe that any $C^1$-map from a compact manifold into…

Let $\mathcal A$ be an $\mathbb F$-algebra and $\omega \in \mathcal A\langle x_1, \ldots, x_m \rangle$ which defines a map $\mathcal A^m \rightarrow \mathcal A$ by evaluation, called a polynomial map with constant. We consider $\mathcal {A}…

We present a framework for characterizing injectivity of classes of maps (on cosets of a linear subspace) by injectivity of classes of matrices. Using our formalism, we characterize injectivity of several classes of maps, including…

For an $n \times n$ matrix $M$ with entries in $\mathbb{Z}_2$ denote by $R(M)$ the minimal rank of all the matrices obtained by changing some numbers on the main diagonal of $M$. We prove that for each non-negative integer $k$ there is a…

Let $\mathbf{O}(\mathbb{F})$ be the split octonion algebra over an algebraically closed field $\mathbb{F}$. For positive integers $k_1, k_2\geq 2$, we study surjectivity of the map $A_1(x^{k_1}) + A_2(y^{k_2}) \in…

Let $G$ be a group containing a nilpotent normal subgroup $N$ with central series $\{N_j\}$, such that each $N_j/N_{j+1}$ is a $\mathbb{F}$-vector space over a field $\mathbb{F}$ and the action of $G$ on $N_j/N_{j+1}$ induced by the…

Let $p$ be a multilinear polynomial in several non-commuting variables with coefficients in a quadratically closed field $K$ of any characteristic. It has been conjectured that for any $n$, the image of $p$ evaluated on the set $M_n(K)$ of…

We give necessary and sufficient conditions in terms of sign vectors for the injectivity of families of polynomial maps with arbitrary real exponents defined on the positive orthant. Our work relates and extends existing injectivity…

We prove in this article the surjectivity of three maps. We prove in Theorem $1.6$ the surjectivity of the Chinese remainder reduction map associated to the projective space of an ideal with a given factorization into ideals whose radicals…

We start from a parametrized system of $d$ generalized polynomial equations (with real exponents) for $d$ positive variables, involving $n$ generalized monomials with $n$ positive parameters. Existence and uniqueness of a solution for all…

For a polynomial map $\mathbf{f} : k^n \to k^m$ ($k$ a field), we investigate those polynomials $g \in k[t_1,\ldots, t_n]$ that can be written as a composition $g = h \circ \mathbf{f}$, where $h: k^m \to k$ is an arbitrary function. In the…

Linear algebra's main concerns are sets of vectors, linear functions, subspaces, linear systems, matrices and concepts about those, such as whether the solution of linear system exists or is unique; a set of vectors is linearly independent…

We give a criterion for maps on ultrametric spaces to be surjective and to preserve spherical completeness. We show how Hensel's Lemma and the multi-dimensional Hensel's Lemma follow from our result. We give an easy proof that the latter…

Let $K$ be an algebraically closed field of characteristic $0$ and let $M_n(K)$, $n \ge 3$, be the matrix ring over $K$. We will show that the image of any multilinear polynomial in four variables evaluated on $M_n(K)$ contains all matrices…

We talk about the image of the Hilbert map. We show the necessary and sufficient condition that the Hilbert map is surjective.

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 present an equivalent statement to the Jacobian conjecture. For a polynomial map F on an affine space of dimension n, we define recursively n finite sequences of polynomials. We give an equivalent condition to the…

Let $w = [[x^k, y^l], [x^m, y^n]]$ be a non-trivial double commutator word. We show that $w$ is surjective on $\mathrm{PSL}_2(K)$, where $K$ is an algebraically closed field of characteristic $0$.