A Note on Polynomial Functions (original) (raw)

[Victor V. Prasolov] Polynomials (Algorithms and C)

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Course 1 ” Polynomials : Their Power and How to Use Them ”

2007

In this paper we observe the problem of counting graph colorings using polynomials. Several reformulations of The Four Color Conjecture are considered (among them algebraic, probabilistic and arithmetic). In the last section Tutte polynomials are mentioned.

On means, polynomials and special functions

We discuss how derivatives can be considered as a game of cubes and beams, and of geometric means. The same principles underlie wide classes of poly-nomials. This results in an unconventional view on the history of the differ-entiation and differentials. In On Proof and Progress in Mathematics William Thurston describes how people develop an "understanding" of mathematics [1]. He uses the exam-1

A new class of polynomial functions equipped with a parameter

Mathematical Sciences, 2017

In this study, a new class of polynomial functions although equipped with a parameter is introduced. This class can be employed for computational solution of linear or non-linear functional equations, including ordinary differential equations or integral equations. The extra parameter permits us to obtain more accurate results. In the present paper, a number of numerical examples show the ability of this class of polynomial functions.

Problems Around Polynomials: The Good, The Bad and The Ugly

Arnold Mathematical Journal, 2015

The Russian style of formulating mathematical problems means that nobody will be able to simplify your formulation as opposed to the French style which means that nobody will be able to generalize it,-Vladimir Arnold.

On a Class of Quadratic Polynomials with No Zeros and Its Applications to APN Functions

2011

We show that the there exists an infinite family of APN functions of the form F (x) = x 2 s +1 + x 2 k+s +2 k + cx 2 k+s +1 + c 2 k x 2 k +2 s + δx 2 k +1 , over F 2 2k , where k is an even integer and gcd(2k, s) = 1, 3 ∤ k. This is actually a proposed APN family of Lilya Budaghyan and Claude Carlet who show in [6] that the function is APN when there exists c such that the polynomial y 2 s +1 + cy 2 s + c 2 k y + 1 = 0 has no solutions in the field F 2 2k. In [6] they demonstrate by computer that such elements c can be found over many fields, particularly when the degree of the field is not divisible by 3. We show that such c exists when k is even and 3 ∤ k (and demonstrate why the k odd case only re-describes an existing family of APN functions). The form of these coefficients is given so that we may write the infinite family of APN functions. APN functions; zeros of polynomials; irreducible polynomials.