相关论文: Density of integer solutions to diagonal quadratic…
Given a linear equation of the form $a_1x_1 + a_2x_2 + a_3x_3 = 0$ with integer coefficients $a_i$, we are interested in maximising the number of solutions to this equation in a set $S \subseteq \mathbb{Z}$, for sets $S$ of a given size. We…
In this paper, we deal with the quartic Diophantine equation $X^4-Y^4=R^2-S^2$ to present its infinitely many integer solutions.
In this paper we investigate the distribution of the set of values of a linear map at integer points on a quadratic surface. In particular we show that this set is dense in the range of the linear map subject to certain algebraic conditions…
The author showed that any homogeneous algebraic Diophantine equation of the second order can be converted to a diagonal form using an integer non-orthogonal transformation maintaining asymptotic behavior of the number of its integer…
An approach is proposed for bounding the number of zeros that solutions of linear differential systems with polynomial coefficients may have. A bound is obtained in a special case which improves upon currently existing.
We give bit-size estimates for the coefficients appearing in triangular sets describing positive-dimensional algebraic sets defined over Q. These estimates are worst case upper bounds; they depend only on the degree and height of the…
Given a prime $p\ge5$ and an integer $s\ge1$, we show that there exists an integer $M$ such that for any quadratic polynomial $f$ with coefficients in the ring of integers modulo $p^s$, such that $f$ is not a square, if a sequence…
We are concerned with the long time behaviour of solutions to the fractional porous medium equation with a variable spatial density. We prove that if the density decays slowly at infinity, then the solution approaches the Barenblatt-type…
An explicit upper bound is established for the least non-trivial integer zero of an arbitrary cubic form $C \in \mathbb{Z}[X_1,...,X_n],$ provided that $n \geq 14.$
We give upper bounds on the size of the gap between a non-zero constant term and the next non-zero Fourier coefficient of an entire level two modular form. We give upper bounds for the minimum positive integer represented by a level two…
We give an asymptotic for the number of prime solutions to $Q(x_1,\dots, x_8) = N$, subject to a mild non-degeneracy condition on the homogeneous quadratic form $Q$. The argument initially proceeds via the circle method, but this does not…
We determine all integers $n$ such that $n^2$ has at most three base-$q$ digits for $q \in \{2, 3, 4, 5, 8, 16 \}$. More generally, we show that all solutions to equations of the shape $$ Y^2 = t^2 + M \cdot q^m + N \cdot q^n, $$ where $q$…
In this note, we give an elementary proof of the following classical fact. Any positive definite ternary quadratic form over the rational numbers fails to represent infinitely many positive integers. For any ternary quadratic form (positive…
We show that for infinitely many square-free integers q there exist infinitely many triples of rational numbers {a, b, c} such that a^2 + q, b^2 + q, c^2 + q, ab + q, ac + q and bc + q are squares of rational numbers.
In this note we consider Diophantine equations of the form \begin{equation*} a(x^p-y^q) = b(z^r-w^s), \quad \mbox{where}\quad \frac{1}{p}+\frac{1}{q}+\frac{1}{r}+\frac{1}{s}=1, \end{equation*} with even positive integers $p,q,r,s$. We show…
Let X be a tight t-design of dimension n for one of the open cases t=5 or t=7. An investigation of the lattice generated by X using arithmetic theory of quadratic forms allows to exclude infinitely many values for n.
We investigate the probability that a random quadratic form in ${\mathbb{Z}}[x_1,...,x_n]$ has a totally isotropic subspace of a given dimension. We show that this global probability is a product of local probabilities. Our main result…
The dual Minkowski problem in the two-dimensional plane is studied in this paper. By combining the theoretical analysis and numerical estimation of an integral with parameters, we find the number of solutions to this problem for the…
We consider a translation and dilation invariant system consisting of k diagonal equations of degrees 1,2,...,k with integer coefficients in s variables, where s is sufficiently large in terms of k. We show via the Hardy-Littlewood circle…
In the paper we partially solved the problem of the distribution of the discriminants of integral polynomials in the cubic case. We proved the asymptotic formula for the number of integral cubic polynomials having bounded height and bounded…