Related papers: On the Jacobian ideal of the binary discriminant

We consider the coincident root loci consisting of the polynomials with at least two double roots andpresent a linear basis of the corresponding ideal in the algebra of symmetric polynomials in terms of the Jack polynomials with special…

Let $R=\oplus_{m\geq 0}R_m$ be a standard graded equidimensional ring over a field $R_0$, and $I\subseteq J$ be two non-nilpotent graded ideals in $R$. Then we give a set of numerical characterizations of the integral dependence of $I$ and…

Let $R$ be a $\mathbb{Q}$-algebra and $d$ be a locally nilpotent derivation on $R$. We will show that the Jacobson radical of a differential polynomial ring $R[x;d]$ equals $I[x;d]$ where $I$ is a nil ideal of $R$. This answers a question…

Let $-D$ be a fundamental discriminant. We express the number of representations of an integer by a positive definite binary quadratic form of discriminant $-D$ with an odd class number $h(-D)$ as a rational linear expression involving the…

Let $p>5$ be a prime. We prove congruences modulo $p^{3-d}$ for sums of the general form $\sum_{k=0}^{(p-3)/2}\binom{2k}{k}t^k/(2k+1)^{d+1}$ and $\sum_{k=1}^{(p-1)/2}\binom{2k}{k}t^k/k^d$ with $d=0,1$. We also consider the special case…

We apply the methods of classical approximation theory (extreme properties of polynomials) to study the essential support $\Sigma_{ac}$ of the absolutely continuous spectrum of Jacobi matrices. First, we prove an upper bound on the measure…

In this paper, the main objective is to compare the abelian subalgebras and ideals of maximal dimension for finite-dimensional supersolvable Lie algebras. We characterise the maximal abelian subalgebras of solvable Lie algebras and study…

We define the uniform face ideal of a simplicial complex with respect to an ordered proper vertex colouring of the complex. This ideal is a monomial ideal which is generally not squarefree. We show that such a monomial ideal has a linear…

We prove that the geometric genus p of a curve in a very generic Jacobian of dimension g>3 satisfies either p=g or p>2g-3. This gives a positive answer to a conjecture of Naranjo and Pirola. For low values of g the second inequality can be…

In this paper we study the (Cohen-Macaulay) type of orders over Dedekind domains in \'etale algebras. We provide a bound for the type, and give formulas to compute it. We relate the type of the overorders of a given order to the size of…

We show that any classical solution of the diffusive Hamilton-Jacobi (DHJ) equation $-\Delta u= |\nabla u|^p$ in a half-space with zero boundary conditions for $1<p\le 2$ is necessarily one-dimensional. This improves the previously known…

In this paper we give a generalization of a result of Herzog, Hibi, and Zheng providing an upper bound for regularity of powers of an ideal. As the main result of the paper, we give a simple criterion in terms of Rees algebra of a given…

We establish a series of results showing that the Jacobian ideal is contained in the test ideal. We first prove a new result in characteristic $p$ for complete rings over a field $K$. Then we prove some results showing that Jacobian ideals…

We prove that for any two integers $g\geq 4$ and $g'\leq 2g-1$, there exist abelian varieties over $\overline{\mathbb{Q}}$ which are not quotients of a Jacobian of dimension $g'$. Our method in fact proves that most Abelian varieties…

It is shown that the deformed Calogero-Moser-Sutherland (CMS) operators can be described as the restrictions on certain affine subvarieties (called generalised discriminants) of the usual CMS operators for infinite number of particles. The…

For a simplicial complex $\Delta$, the affect of the expansion functor on combinatorial properties of $\Delta$ and algebraic properties of its Stanley-Reisner ring has been studied in some previous papers. In this paper, we consider the…

For redundant second-class constraints the Dirac brackets cannot be defined and new brackets must be introduced. We prove here that the Jacobi identity for the new brackets must hold on the surface of the second-class constraints. In order…

In the first part of this paper we study scrollers and linearly joined varieties. A particular class of varieties, of important interest in classical Geometry are Cohen--Macaulay varieties of minimal degree. They appear naturally studying…

We describe a qualitative improvement to the algorithms for performing 2-descents to obtain information regarding the Mordell-Weil rank of a hyperelliptic Jacobian. The improvement has been implemented in the Magma Computational Algebra…

We study codimension two determinantal varieties with isolated singularities. These singularities admit a unique smoothing, thus we can define their Milnor number as the middle Betti number of their generic fiber. For surfaces in C^4, we…