Related papers: Least zero of a cubic form
One of the fundamental problem in the theory of sorting is to find the pessimistic number of comparisons sufficient to sort a given number of elements. Currently 16 is the lowest number of elements for which we do not know the exact value.…
We study four (families of) sets of algebraic integers of degree less than or equal to three. Apart from being simply defined, we show that they share two distinctive characteristics: almost uniformity and arithmetical independence. Here,…
Assuming the Generalized Riemann Hypothesis, the non-trivial zeros of $L$-functions lie on the critical line with the real part $1/2$. We find an upper bound of the lowest first zero in families of even cuspidal newforms of prime level…
If $\mathscr A$ is a set of natural numbers of exponential density $\delta$, then the exponential density of all numbers of the form $x^3+a$ with $x\in\mathbb N$ and $a\in\mathscr A$ is at least $\min(1, \frac 13+\frac 56 \delta)$. This is…
In this article, we derive lower bounds for the number of distinct prime divisors of families of non-zero Fourier coefficients of non-CM primitive cusp forms and more generally of non-CM primitive Hilbert cusp forms. In particular, for the…
Approximate integer programming is the following: For a convex body $K \subseteq \mathbb{R}^n$, either determine whether $K \cap \mathbb{Z}^n$ is empty, or find an integer point in the convex body scaled by $2$ from its center of gravity…
We present a practical algorithm to compute models of rational functions with minimal resultant under conjugation by fractional linear transformations. We also report on a search for rational functions of degrees 2 and 3 with rational…
Given a finite set of arbitrarily distributed points in affine space with arbitrary multiplicity structures, we present an algorithm to compute the reduced Groebner basis of the vanishing ideal under the lexicographic ordering. Our method…
In this paper, we study the problem of computing a minimum-width axis-aligned cubic shell that encloses a given set of $n$ points in a three-dimensional space. A cubic shell is a closed volume between two concentric and face-parallel cubes.…
We study the lexicographically least infinite $a/b$-power-free word on the alphabet of non-negative integers. Frequently this word is a fixed point of a uniform morphism, or closely related to one. For example, the lexicographically least…
Some new bounds for the extreme zeroes of Jacobi polynomials are obtained with an elementary approach. A feature of these bounds is their simple forms, which make them easy to work with. Despite their simplicity, our lower bounds for the…
The paper covers a formulation of the inverse quadratic programming problem in terms of unconstrained optimization where it is required to find the unknown parameters (the matrix of the quadratic form and the vector of the quasi-linear part…
We prove that an additive form of degree $d=2m$, $m$ odd, $m\ge3$, over the unramified quadratic extension $\mathbb{Q}_2(\sqrt{5})$ has a nontrivial zero if the number of variables $s$ satisifies $s \ge 4d+1$. If $3 \nmid d$, then there…
We study the minimum number of minimal codewords in linear codes from the point of view of projective geometry. We derive bounds and in some cases determine the exact values. We also present an extension to minimal subcode supports.
Some geometry on non-singular cubic curves, mainly over finite fields, is surveyed. Such a curve has 9,3,1 or 0 points of inflexion, and cubic curves are classified accordingly. The group structure and the possible numbers of rational…
We study the problem of whether a given finite algebra with finitely many basic operations contains a cube term; we give both structural and algorithmic results. We show that if such an algebra has a cube term then it has a cube term of…
It is conjectured that every integer N>454 is the sum of seven nonnegative cubes. We prove the conjecture when N is a multiple of 4.
The so-called {\it zero number diminishing property} (or {\it zero number argument}) is a powerful tool in qualitative studies of one dimensional parabolic equations, which says that, under the zero- or non-zero-Dirichlet boundary…
A perfect cuboid is a rectangular parallelepiped with integer edges and integer face diagonals whose space diagonal is also integer. The existence of such cuboids is neither proved, nor disproved. A rational perfect cuboid is a natural…
Cubic surfaces in characteristic two are investigated from the point of view of prime characteristic commutative algebra. In particular, we prove that, the non-Frobenius split cubic surfaces form a linear subspace of codimension four in the…