Related papers: On the geometry of zero sets of central quaternion…

Let G be a graph obtained by taking r>=2 paths and identifying all first vertices and identifying all the last vertices. We compute the Castelnuovo--Mumford regularity of the quotient S/I(X), where S is the polynomial ring on the edges of G…

In this paper we prove a quaternionic positive real lemma as well as its generalized version, in case the associated kernel has negative squares for slice hyperholomorphic functions. We consider the case of functions with positive real part…

In this paper, we derive explicit formulas for computing the roots of $ax^{2}+bx+c=0$ with $a$ being not invertible in split quaternion algebra. We also imitate the approach developed by Opfer, Janovska and Falcao etc. to verify our results…

Let $R$ be a smooth affine algebra over an infinite perfect field $k$. Let $I\subset R$ be an ideal, $\omega_I:(R/I)^n\to I/I^2$ a surjective homomorphism and $Q_{2n}\subset \mathbb{A}^{2n+1}$ be the smooth quadric defined by the equation…

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…

We use a method based on the division algorithm to determine all the values of the real parameters $b$ and $c$ for which the hypergeometric polynomials $_2F_1(-n, b; c; z)$ have $n$ real, simple zeros. Furthermore, we use the…

Given a central division algebra $D$ of degree $d$ over a field $F$, we associate to any standard polynomial $\phi(z)=z^n+c_{n-1} z^{n-1}+\dots+c_0$ over $D$ a "companion polynomial" $\Phi(z)$ of degree $n d$ with coefficients in $F$ whose…

Let $R$ be a ring with ${\bf 1}$ which is not commutative. Assume that a non-zero commutator in $R$ is not a zero divisor. Assume further that either $R$ is alternative, but not associative, or $R$ is associative and any commutator $v\in R$…

It is an open problem to determine the dimension of the space of homogeneous polynomials of a fixed degree vanishing at finitely many points in the projective plane to certain multiplicities. We present various aspects of this problem and a…

Fix an integer $n \geq 1$, and consider the set of all connected finite simple graphs on $n$ vertices. For each $G$ in this set, let $I(G)$ denote the edge ideal of $G$ in the polynomial ring $R = K[x_1,\ldots,x_n]$. We initiate a study of…

Given an embedded smooth projective variety Y in CP^n, we show how the existence of a hypersurface with high multiplicity along Y, but of relatively low degree and log canonical near Y implies vanishing of higher cohomology for certain…

It is well known that if one integrates a Schur function indexed by a partition $\lambda$ over the symplectic (resp. orthogonal) group, the integral vanishes unless all parts of $\lambda$ have even multiplicity (resp. all parts of $\lambda$…

Assuming an integral quadratic polynomial with nonsingular quadratic part has a nontrivial zero on an integer lattice outside of a union of finite-index sublattices, we prove that there exists such a zero of bounded norm and provide an…

The symbolic powers $I^{(n)}$ of a radical ideal $I$ in a polynomial ring consist of the functions that vanish up to order $n$ in the variety defined by $I$. These do not necessarily coincide with the ordinary algebraic powers $I^n$, but it…

We show that a ring $R$ is regular if $Tor_{i}^{R}(R^{+},k) = 0$ for some $i\geq 1$ assuming further that $R$ is a $\mathbb{N}$-graded ring of dimension $2$ finitely generated over an equi-characteristic zero field $k$. This answers a…

Let $I$ be a two-dimensional squarefree monomial ideal of a polynomial ring $S$. We evaluate the geometric regularity, $a_i$-invariants for $i\geq 1$ of the power $I^n$. It turns out they are all linear functions in $n$ from $n=2$.…

Let (M,I,J,K) be a hyperkahler manifold of real dimension 4n, and L a non-trivial holomorphic line bundle on (M,I). Using the quaternionic Dolbeault complex, we prove the following vanishing theorem for holomorphic cohomology of L. If the…

The decomposition of the polynomials on the quaternionic unit sphere in $\Hd$ into irreducible modules under the action of the quaternionic unitary (symplectic) group and quaternionic scalar multiplication has been studied by several…

Assume M is a 3-dimensional real manifold without boundary, A is an abelian Lie algebra of analytic vector fields on M, and X is an element of A. The following result is proved: If K is a locally maximal compact set of zeroes of X and the…

A physically more adequate definition of a quaternionic holomorphic (H-holomorphic) function of one quaternionic variable compared to known ones and a quaternionic generalization of Cauchy-Riemann's equations are presented. At that a class…