Related papers: Polynomial representation of Fermat's Last Theorem
In this paper we mainly study the existence and the form of entire solutions with finite order for the following system of Fermat-type difference and partial differential-difference equations $$\begin{cases} f_1(z)^2+(\Delta_cf_2(z))^2=1\cr…
Univariate polynomial root-finding is a classical subject, still important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the…
We give a short proof of the well-known Knuth's old sum and provide some generalizations. Our approach utilizes the binomial theorem and integration formulas derived using the Beta function. Several new polynomial identities and…
We present the finite first-order theory (FFOT) machine, which provides an atemporal description of computation. We then develop a concept of complexity for the FFOT machine, and prove that the class of problems decidable by a FFOT machine…
We develop the Perron-Frobenius theory using a variational approach and extend it to a set of arbitrary matrices, including those that are neither irreducible nor essentially positive, and non-preserved cones. We introduce a new concept…
Carlitz proved that, for any prime power q other than 2, the group of all permutations of the finite field F_q is generated by the permutations induced by degree-one polynomials and x^{q-2}. His proof relies on a remarkable polynomial which…
We develop a new tool, namely polynomial and linear algebraic methods, for studying systems of word equations. We illustrate its usefulness by giving essentially simpler proofs of several hard problems. At the same time we prove extensions…
We investigate Newton's method as a root finder for complex polynomials of arbitrary degree. While polynomial root finding continues to be one of the fundamental tasks of computing, with essential use in all areas of theoretical…
We present a general technique for obtaining permutation polynomials over a finite field from permutations of a subfield. By applying this technique to the simplest classes of permutation polynomials on the subfield, we obtain several new…
The bivariate difference filed $(\mathbb{F}(\alpha, \beta), \sigma)$ provides an algebraic framework for a sequence satisfying a recurrence of order two and it could transform the summation involving a sequence satisfying a recurrence of…
This paper reviews the current state of the art of the mean value theorem due to Thomas M. Flett. We present the results with detailed proofs and provide many new proofs of known results. Moreover, some new observations and yet unpublished…
The probability that a zero of a random real polynomial of increasing degree is real tends to zero. However, passing from polynomials to Laurent polynomials yields a surprising result: the probability that a root is real tends not to zero,…
We give an especially simple proof of a theorem in graph theory that forms the key part of the solution to a problem in commutative algebra, on how to characterize the integral closure of a polynomial ring generated by quadratic monomials.
This expository essay discusses a finite dimensional approach to dilation theory. How much of dilation theory can be worked out within the realm of linear algebra? It turns out that some interesting and simple results can be obtained. These…
We present a complete algorithm for finding an exact minimal polynomial from its approximate value by using an improved parameterized integer relation construction method. Our result is superior to the existence of error controlling on…
Though it is well known that the roots of any affine polynomial over a finite field can be computed by a system of linear equations by using a normal base of the field, such solving approach appears to be difficult to apply when the field…
Polynomial factorization and root finding are among the most standard themes of computational mathematics. Yet still, little has been done for polynomials over quaternion algebras, with the single exception of Hamiltonian quaternions for…
We will show that the roots of a polynomial equation in one variable of degree n are related to the solutions of a symmetric quadratic form in n-1 variables with constant positive integer coefficients. The classic polynomial notation will…
Comments about the paper by Elsholz, Fermat's last theorem implies Euclid's infinitude of primes, (2021), and simplification.
Pellet's theorem determines when the zeros of a polynomial can be separated into two regions, based on the presence or absence of positive roots of an auxiliary polynomial, but does not provide a method to verify its conditions or to…