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Let $g(X)$ be a polynomial over a finite field ${\mathbb F}_q$ with degree $o(q^{1/2})$, and let $\chi$ be the quadratic residue character. We give a polynomial time algorithm to recover $g(X)$ (up to perfect square factors) given the…

For prime $p$ and small $n$, Jones and Roberts have developed a database recording invariants for $p$-adic extensions of degree $n$. We contributed to this database by computing the Galois slope content, Galois mean slope, and inertia…

The fastest known algorithm for factoring univariate polynomials over finite fields is the Kedlaya-Umans (fast modular composition) implementation of the Kaltofen-Shoup algorithm. It is randomized and takes $\widetilde{O}(n^{3/2}\log q + n…

We show a new algorithm and its implementation for multiplying bit-polynomials of large degrees. The algorithm is based on evaluating polynomials at a specific set comprising a natural set for evaluation with additive FFT and a high order…

Modular algorithm are widely used in computer algebra systems (CAS), for example to compute efficiently the gcd of multivariate polynomials. It is known to work to compute Groebner basis over $\Q$, but it does not seem to be popular among…

Suppose f is a real univariate polynomial of degree D with exactly 4 monomial terms. We present an algorithm, with complexity polynomial in log D on average (relative to the stable log-uniform measure), for counting the number of real roots…

The classical division algorithm for polynomials requires $O(n^2)$ operations for inputs of size $n$. Using reversal technique and Newton iteration, it can be improved to $O({M}(n))$, where ${M}$ is a multiplication time. But the method…

It is common in stability analysis to linearize a system and investigate the spectrum of the Jacobian matrix. This approach faces the challenge of determining the matrix spectrum when the coefficients depend on parameters or when the…

Expansive polynomials (whose roots are greater than 1 in modulus) often arise in dynamical systems and other computational problems. This paper examines the expansivity gap (the gap between 1 and the smallest modulus of the roots) of these…

We give two determinantal representations for a bivariate polynomial. They may be used to compute the zeros of a system of two of these polynomials via the eigenvalues of a two-parameter eigenvalue problem. The first determinantal…

In a recent paper Lima, Panario and Wang have provided a new method to multiply polynomials in Chebyshev basis which aims at reducing the total number of multiplication when polynomials have small degree. Their idea is to use Karatsuba's…

We have been working in many aspects of the problem of analyzing, understanding and solving ordinary differential equations (first and second order). As we have extensively mentioned, while working in the Darboux type methods, the most…

In this paper, we present a new basis of polynomial over finite fields of characteristic two and then apply it to the encoding/decoding of Reed-Solomon erasure codes. The proposed polynomial basis allows that $h$-point polynomial evaluation…

To every integer monic polynomial of degree m can be associated a `replacement rule' that generates a word W* from another word W consisting of symbols belonging to a finite `alphabet' of size 2m. This rule applied iteratively on almost any…

We revisit the notion of root polynomials, thoroughly studied in [F. Dopico and V. Noferini, Root polynomials and their role in the theory of matrix polynomials, Linear Algebra Appl. 584:37--78, 2020] for general polynomial matrices, and…

Higher-order mutation has the potential for improving major drawbacks of traditional first-order mutation, such as by simulating more realistic faults or improving test optimization techniques. Despite interest in studying promising…

We improve the local generic position method for isolating the real roots of a zero-dimensional bivariate polynomial system with two polynomials and extend the method to general zero-dimensional polynomial systems. The method mainly…

This is a book about computational aspects of modular forms and the Galois representations attached to them. The main result is the following: Galois representations over finite fields attached to modular forms of level one can, in almost…

For interior-point algorithms in linear programming, it is well-known that the selection of the centering parameter is crucial for proving polynomility in theory and for efficiency in practice. However, the selection of the centering…

We propose a general methodology for testing whether a given polynomial with integer coefficients is identically zero. The methodology evaluates the polynomial at efficiently computable approximations of suitable irrational points. In…