Related papers: Solving conics over Q(t1,..,tk)
We construct an algorithm for solving the following problem: given a number field $K$, a positive integer $N$, and a positive real number $B$, determine all points in $\mathbb P^N(K)$ having relative height at most $B$. A theoretical…
Computing the real solutions to a system of polynomial equations is a challenging problem, particularly verifying that all solutions have been computed. We describe an approach that combines numerical algebraic geometry and sums of squares…
The Tijdeman-Zagier conjecture states no integer solution exists for $A^X+B^Y=C^Z$ with positive integer bases and integer exponents greater than 2 unless gcd$(A,B,C)>1$. Any set of values that satisfy the conjecture correspond to a lattice…
The conditions for cubic equations, to have 3 real roots and 2 of the roots lie in the closed interval $[-1, 1]$ are given. These conditions are visualized. This question arises in physics in e.g. the theory of tops.
Considering the equation $$a^x+b^y = c^z,$$ Le gives an upper bound on the number of solutions $(x,y,z)$, and a bound on $z$. Here we give slight improvements to these results, also removing the restriction that $a$, $b$, $c$ be squarefree.
This paper deals with the numerical computation of the least singular value of a rectangular matrix $A$ relative to a pair of closed convex cones $(P,Q)$, which is defined as the optimal value of the non-convex optimization problem of…
We show that if we assume Martin's Axiom, then there exists a nontrivial twisted sum of c_0 and C(K), for every compact space K with finite height and weight at least continuum. This result settles the problem of existence of nontrivial…
We study solutions of a homogeneous quadratic equation $q(x_0,\dots, x_n)=0$, defined over a field $K$, where the $x_i$ are themselves homogeneous polynomials of some degree $d$ in $r+1$ variables. Equivalently, we are looking at rational…
We develop a systematic way to solve linear equations involving tensors of arbitrary rank. We start off with the case of a rank $3$ tensor, which appears in many applications, and after finding the condition for a unique solution we derive…
A lattice point in $\mathbb{R}^2$ is a point $(x,y)$ with $x,y\in\mathbb{Z}$, and a lattice triangle is a triangle whose three vertices are all lattice points. We investigate the integers $k$ with the property that if $T$ is a lattice…
A Chebyshev curve C(a,b,c,\phi) has a parametrization of the form x(t)=Ta(t); y(t)=T_b(t) ; z(t)= Tc(t + \phi), where a,b,c are integers, Tn(t) is the Chebyshev polynomial of degree n and \phi \in \RR. When C(a,b,c,\phi) has no double…
In this exposition-type note we present detailed proofs of certain assertions concerning several algebraic properties of the cone and cylinder algebras. These include a determination of the maximal ideals, the solution of the B\'ezout…
A simple algorithm to compute all the zeros of a generic polynomial is proposed.
We study the proportion of conics given by $(\mathcal{C}_{\mathbf{F}, \mathbf{y}}) : F_0(\mathbf{y})x_0^2 + F_1(\mathbf{y})x_1^2 = F_2( \mathbf{y})x_2^2 $ which have a rational point $\mathbf{x} = (x_0 :x_1:x_2) \in…
We count the number of conics through two general points in complete intersections when this number is finite and give an application in terms of quasi-lines.
In this paper we develop a new theory for the existence, localization and multiplicity of positive solutions for a class of non-variational,quasilinear, elliptic systems. In order to do this, we provide a fairly general abstract framework…
In [5], Srijuntongsiri and Vavasis propose the "Kantorovich-Test Subdivision algorithm", or KTS, which is an algorithm for finding all zeros of a polynomial system in a bounded region of the plane. This algorithm can be used to find the…
When $A=3$, the positive integral solutions of the so-called Markoff equation $$M_A:x^2 + y^2 + z^2 = Axyz$$ can be generated from the single solution $(1,1,1)$ by the action of certain automorphisms of the hypersurface. Since Markoff's…
In this paper we completely solve a simple quartic family of Thue equations over $\mathbb{C}(T)$. Specifically, we apply the ABC-Theorem to find all solutions $(x,y) \in \mathbb{C}[T] \times \mathbb{C}[T]$ to the set of Thue equations…
Consider the $n$th degree polynomial equation, $X^n+A_{n-1}X^{n-1}+...+A_1X+A_0=0$ over the ring of 2 by 2 complex matrices. If this equation has more than ${2n \choose 2}$ solutions, then it has infinitely many solutions. We show here that…