Related papers: Cubefree Trinomial Discriminants

We determine the probability that a random n x n symmetric matrix over {1, 2, ... , m} has determinant divisible by m.

The divisibility of truncated binomial series by their exponent n is analyzed. Divisibility is shown to depends on the divisibility characteristics of the integers constituting the binomials. Series division by the highest possible powers…

The number of non-isomorphic cubic fields L sharing a common discriminant d(L) = d is called the multiplicity m = m(d) of d. For an assigned value of d, these fields are collected in a multiplet M(d) = (L(1) ,..., L(m)). In this paper, the…

In this paper, we prove a theorem on the distribution of primes in cubic progressions on average.

We prove large sieve inequalities with multivariate polynomial moduli and deduce a general Bombieri--Vinogradov type theorem for a class of polynomial moduli having a sufficient number of variables compared to its degree. This sharpens…

In a recent paper almost sure unisolvence of RBF interpolation at random points with no polynomial addition was proved, for Thin-Plate Splines and Radial Powers with noninteger exponent. The proving technique left unsolved the case of odd…

Primitive prime divisors play an important role in group theory and number theory. We study a certain number theoretic quantity, called $\Phi^*_n(q)$, which is closely related to the cyclotomic polynomial $\Phi_n(x)$ and to primitive prime…

Let ${\{K_m\}_{m\geq 4}}$ be the family of non-normal totally real cubic number fields defined by the irreducible cubic polynomial $f_m(x)=x^3-mx^2-(m+1)x-1$, where $m$ is an integer with $m\geq 4$. In this paper, we will give a class…

We establish asymptotic formulae for the number of $k$-free values of polynmilas $F(x_1,\cdots,x_n)\in\mathbb{Z}[x_1,\cdots,x_n]$ of degree $d\geq 2$ for any $n\geq 1$, including when the variables are prime, as long as $k\geq (3d+1)/4$.…

A class P_{n,m,p}(x) of polynomials is defined. The combinatorial meaning of its coefficients is given. Chebyshev polynomials are the special cases of P_{n,m,p}(x). It is first shown that P_{n,m,p}(x) may be expressed in terms of…

We show under a mild hypothesis that given field elements $a_0, \dots, a_m \in K$, there always exists a degree-$m$ polynomial whose $n$th power whose degree-$jn$ coefficient is equal to $a_j$ for $0 \leq j \leq m$. We provide an alternate…

It is well-known that for any non-constant polynomial $P$ with integer coefficients the sequence $(P(n))_{ n\in \mathbb N}$ has the property that there are infinitely many prime numbers dividing at least one term of this sequence.…

Let $s_{ij}$ represent a tranposition in $S_n$. A polynomial $P$ in $\mathbb{Q}[X_n]$ is said to be $m$-quasiinvariant with respect to $S_n$ if $(x_i-x_j)^{2m+1}$ divides $(1-s_{ij})P$ for all $1 \leq i, j \leq n$. We call the ring…

We prove that any pair of bivariate trinomials has at most 5 isolated roots in the positive quadrant. The best previous upper bounds independent of the polynomial degrees were much larger, e.g., 248832 (for just the non-degenerate roots)…

Let $\mathbf{P}^{m}_{b}(x)$ be a $2m+1$-degree polynomial in $x$ and $b \in \mathbb{R}$ \[ \mathbf{P}^{m}_{b}(x) = \sum_{k=0}^{b-1} \sum_{r=0}^{m} \mathbf{A}_{m,r} k^r (x-k)^r \] where $\mathbf{A}_{m,r}$ are real coefficients. In this…

In this paper we find the number of homogeneous polynomials of degree d such that they vanish on cuspidal modular forms of even weight $m\geq 2$ that form a basis for $S_m(\Gamma_0(N))$. We use these cuspidal forms to embedd $X_0(N)$ to…

We study the irreducibility of Wronskian Hermite polynomials labelled by partitions. It is known that these polynomials factor as a power of x times a remainder polynomial. We show that the remainder polynomial is irreducible for the…

We develop a method for determining the density of squarefree values taken by certain multivariate integer polynomials that are invariants for the action of an algebraic group on a vector space. The method is shown to apply to the…

Let $f \in \mathbb{R}[x]$ be a polynomial with real coefficients. We say that $f$ is eventually non-negative if $f^m$ has non-negative coefficients for all sufficiently large $m \in \mathbb{N}$. In this short note, we give a classification…

We investigate the average distribution of primes represented by positive definite integral binary quadratic forms, the average being taken over negative fundamental discriminants in long ranges. In particular, we prove corresponding…