Related papers: Polynomial Detection of Matrix Subalgebras

In this paper we describe the polynomial identities of degree 4 for a certain subspace of the Weyl algebra A_1 over an infinite field of arbitrary characteristic.

Given two polynomials f(x) and g(x), we extend the formula expressing the remainder in terms of the roots of these two polynomials to the case where f(x) is a Laurent polynomial. This allows us to give new expressions of a Schur function,…

In [19], Taslaman, Tisseur, and Zaballa show that any regular matrix polynomial $P(\lambda)$ over an algebraically closed field is spectrally equivalent to a triangular matrix polynomial of the same degree. When $P(\lambda)$ is real and…

The algebra of $n\times n$ matrices over a field $F$ has a natural $\mathbb{Z}_n$-grading. Its graded identities have been described by Vasilovsky who extended a previous work of Di Vincenzo for the algebra of $2\times 2$ matrices. In this…

We equip the complex polynomial algebra C[t] with the involution which is the identity on C and sends t to -t. Answering a question raised by V.G. Kac, we show that every hermitian or skew-hermitian matrix over this algebra is congruent to…

Some known results for locating the roots of polynomials are extended to the case of matrix polynomials. In particular, a theorem by A.E. Pellet [Bulletin des Sciences Math\'ematiques, (2), vol 5 (1881), pp.393-395], some results of D.A.…

Consider an abstract operator $L$ which acts on monomials $x^n$ according to $L x^n= \lambda_n x^n + \nu_n x^{n-2}$ for $\lambda_n$ and $\nu_n$ some coefficients. Let $P_n(x)$ be eigenpolynomials of degree $n$ of $L$: $L P_n(x) = \lambda_n…

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 prove polynomial identities for the N=1 superconformal model SM(2,4\nu) which generalize and extend the known Fermi/Bose character identities. Our proof uses the q-trinomial coefficients of Andrews and Baxter on the bosonic side and a…

Let $K$ be any field with $\textup{char}K\neq 2,3$. We classify all cubic homogeneous polynomial maps $H$ over $K$ with $\textup{rk} JH\leq 2$. In particular, we show that, for such an $H$, if $F=x+H$ is a Keller map then $F$ is invertible,…

We determine the multiplicity of the irreducible representation V(n) of the simple Lie algebra sl(2,C) as a direct summand of its fourth exterior power $\Lambda^4 V(n)$. The multiplicity is 1 (resp. 2) if and only if n = 4, 6 (resp. n = 8,…

Let $\cal{A}$ be the algebra of quaternions $\mathbb{H}$ or octonions $\mathbb{O}$. In this manuscript a new proof is given, based on ideas of Cauchy and D' Alembert, of the fact that an ordinary polynomial $f(t) \in {\cal{A}}\, [t]$ has a…

Let $\mathbb F$ be a field $\mathbb F $ of characteristic zero. Let $W_{n}(\mathbb F)$ be the Lie algebra of all $\mathbb F$-derivations with the Lie bracket $[D_1, D_2]:=D_1D_2-D_2D_1$ on the polynomial ring $\mathbb F [x_1, \ldots ,…

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…

Let $(V,\omega)$ be an orthosympectic $\mathbb Z_2$-graded vector space and let $\mathfrak g:=\mathfrak{gosp}(V,\omega)$ denote the Lie superalgebra of similitudes of $(V,\omega)$. When the space $\mathscr P(V)$ of superpolynomials on $V$…

We prove a robust generalization of a Sylvester-Gallai type theorem for quadratic polynomials, generalizing the result in [S'20]. More precisely, given a parameter $0 < \delta \leq 1$ and a finite collection $\mathcal{F}$ of irreducible and…

We study the differential identities of the algebra $M_k(F)$ of $k\times k$ matrices over a field $F$ of characteristic zero when its full Lie algebra of derivations, $L=\mbox{Der}(M_k(F))$, acts on it. We determine a set of 2 generators of…

S. Cappell and J. Shaneson constructed a pair of inequivalent embeddings of $(n-1)$-spheres in homotopy $(n+1)$-spheres for every square matrix of order $n$ with special properties (a Cappell-Shaneson matrix). A Cappell-Shaneson polynomial…

We construct, for every even dimensional sphere $S^n$, $n >1$, and every odd integer $k$, a homogeneous polynomial map $f: S^{n}\to S^{n}$ of Brouwer degree $k$ and algebraic degree $2|k|-1$.

We produce a new family of polynomials f(x) over fields K of characteristic 2 which are exceptional, in the sense that f(x)-f(y) has no absolutely irreducible factors in K[x,y] besides the scalar multiples of x-y; when K is finite, this…