Related papers: Estimating heights using auxiliary functions
We consider the problem of constructing semisimple subalgebras of real (semi-) simple Lie algebras. We develop computational methods that help to deal with this problem. Our methods boil down to solving a set of polynomial equations. In…
We present a novel numerical method for solving ODEs while preserving polynomial first integrals. The method is based on introducing multiple quadratic auxiliary variables to reformulate the ODE as an equivalent but higher-dimensional ODE…
We introduce the classes of descendingly flexible and descendingly alternative algebras over an arbitrary field $\mathbb{F}$. We suggest a new method based on the sequence of differences between the dimensions of the linear spans of words,…
Starting from Rodrigues formula we present a general construction of raising and lowering operators for orthogonal polynomials of continuous and discrete variable on uniform lattice. In order to have these operators mutually adjoint we…
We provide bounds on the size of polynomial differential equations obtained by executing closure properties for D-algebraic functions. While it is easy to obtain bounds on the order of these equations, it requires some more work to derive…
Multivariate piecewise polynomial functions (or splines) on polyhedral complexes have been extensively studied over the past decades and find applications in diverse areas of applied mathematics including numerical analysis, approximation…
Let $K$ be a number field, and let $W$ be a subspace of $K^N$, $N \geq 1$. Let $V_1,...,V_M$ be subspaces of $K^N$ of dimension less than dimension of $W$. We prove the existence of a point of small height in $W \setminus \bigcup_{i=1}^M…
The auxiliary field method is a new and efficient way to compute approximate analytical eigenenergies and eigenvectors of the Schr\"{o}dinger equation. This method has already been successfully applied to the case of central potentials of…
When using images to locate objects, there is the problem of correcting for distortion and misalignment in the images. An elegant way of solving this problem is to generate an error correcting function that maps points in an image to their…
The paper deals with the {\it infinitesimal Hilbert 16th problem}: to find an upper estimate of the number of zeros of an Abelian integral regarded as a function of a parameter. In more details, consider a real polynomial $ H$ of degree $…
Let $K$ be a number field, $\overline{\mathbb Q}$, or the field of rational functions on a smooth projective curve over a perfect field, and let $V$ be a subspace of $K^N$, $N \geq 2$. Let $Z_K$ be a union of varieties defined over $K$ such…
Motivated by numerical methods for solving parametric partial differential equations, this paper studies the approximation of multivariate analytic functions by algebraic polynomials. We introduce various anisotropic model classes based on…
We introduce a method to estimate sums of oscillating functions on finite abelian groups over intervals or (generalized) arithmetic progressions, when the size of the interval is such that the completing techniques of Fourier analysis are…
We derive inclusion regions for the eigenvalues of matrix polynomials expressed in a general polynomial basis, which can lead to significantly better results than traditional bounds. We present several applications to engineering problems.
We discuss a new method to compute the canonical height of an algebraic point on a hyperelliptic jacobian over a number field. The method does not require any geometrical models, neither $p$-adic nor complex analytic ones. In the case of…
We obtain a small improvement of Gallagher's larger sieve and we extend it to higher dimensions. We also obtain two interesting upper bounds for the number of solutions to polynomial congruences.
We give an elementary introduction to some recent polyhedral techniques for understanding and solving systems of multivariate polynomial equations. We provide numerous concrete examples and illustrations, and assume no background in…
In these short notes, we will show the following. Let F_q be a finite field and let E/\F_q be an elliptic curve. Let S_r be the rth summation/Semaev polynomial for E. Under an assumption, we show that it is NP-complete to check if S_r…
In this note we give a theoretical support by means of quotient polynomial rings for the computation formulas of the dimension of abelian codes.
We discuss methods for using the Weil polynomial of an isogeny class of abelian varieties over a finite field to determine properties of the curves (if any) whose Jacobians lie in the isogeny class. Some methods are strong enough to show…