## Complete permutation polynomials of monomial type

Some generalizations of known permutation binomials and trinomials that involve a sort of symmetric polynomials are given. Other constructions are based on the study of algebraic curves associated to certain polynomials. This is a preview of subscription content, log in to check access. Rent this article via DeepDyve. Akbary A.

## Some new results on permutation polynomials over finite fields

Bartoli D. Finite Fields Appl. Number Theory46—66 Ding C. SIAM J. Gupta R. Hou X. Acta Arith. Laigle-Chapuy Y. Li, N. Li, K. Lidl R. Cambridge University Press, Cambridge Google Scholar. Marcos J. Mullen G. Park Y. Yuan P. Zhou Y. Zieve M. Zieve, M. In: Additive Number Theory, pp. Download references. We would like to thank the referees for providing us with useful comments which served to improve the paper.

Correspondence to Luciane Quoos. Reprints and Permissions. Bartoli, D. Codes Cryptogr. Download citation. Received : 13 February Enter expression, e. Enter a set of expressions, e.

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## Symmetric polynomial

Enter equation to graph, e. Number of equations to solve: 2 3 4 5 6 7 8 9 Sample Problem Equ. Enter inequality to solve, e.

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### Multiplying Monomials Worksheets

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Back to top. Sample Problem. Find GCF. Find LCM. Depdendent Variable. Number of equations to solve:. Solve for:. Auto Fill. Dependent Variable. Number of inequalities to solve:. Your Website:. Posted: Friday 29th of Dec I'm getting really bored in my math class.In mathematicsa symmetric polynomial is a polynomial P X 1X 2…, X n in n variables, such that if any of the variables are interchanged, one obtains the same polynomial.

Symmetric polynomials arise naturally in the study of the relation between the roots of a polynomial in one variable and its coefficients, since the coefficients can be given by polynomial expressions in the roots, and all roots play a similar role in this setting. From this point of view the elementary symmetric polynomials are the most fundamental symmetric polynomials. A theorem states that any symmetric polynomial can be expressed in terms of elementary symmetric polynomials, which implies that every symmetric polynomial expression in the roots of a monic polynomial can alternatively be given as a polynomial expression in the coefficients of the polynomial.

Symmetric polynomials also form an interesting structure by themselves, independently of any relation to the roots of a polynomial. In this context other collections of specific symmetric polynomials, such as complete homogeneouspower sumand Schur polynomials play important roles alongside the elementary ones.

The resulting structures, and in particular the ring of symmetric functionsare of great importance in combinatorics and in representation theory.

The following polynomials in two variables X 1 and X 2 are symmetric:. There are many ways to make specific symmetric polynomials in any number of variables see the various types below.

An example of a somewhat different flavor is. Similarly in three variables. However, the following is symmetric:. One context in which symmetric polynomial functions occur is in the study of monic univariate polynomials of degree n having n roots in a given field.

These n roots determine the polynomial, and when they are considered as independent variables, the coefficients of the polynomial are symmetric polynomial functions of the roots. Moreover the fundamental theorem of symmetric polynomials implies that a polynomial function f of the n roots can be expressed as another polynomial function of the coefficients of the polynomial determined by the roots if and only if f is given by a symmetric polynomial.

This yields the approach to solving polynomial equations by inverting this map, "breaking" the symmetry — given the coefficients of the polynomial the elementary symmetric polynomials in the rootshow can one recover the roots? This leads to studying solutions of polynomials using the permutation group of the roots, originally in the form of Lagrange resolventslater developed in Galois theory. There exist n roots x 1 ,…, x n of P in some possibly larger field for instance if k is the field of real numbersthe roots will exist in the field of complex numbers ; some of the roots might be equal, but the fact that one has all roots is expressed by the relation.

Now one may change the point of view, by taking the roots rather than the coefficients as basic parameters for describing Pand considering them as indeterminates rather than as constants in an appropriate field; the coefficients a i then become just the particular symmetric polynomials given by the above equations.

A basic fact, known as the fundamental theorem of symmetric polynomials states that any symmetric polynomial in n variables can be given by a polynomial expression in terms of these elementary symmetric polynomials.

It follows that any symmetric polynomial expression in the roots of a monic polynomial can be expressed as a polynomial in the coefficients of the polynomial, and in particular that its value lies in the base field k that contains those coefficients. Thus, when working only with such symmetric polynomial expressions in the roots, it is unnecessary to know anything particular about those roots, or to compute in any larger field than k in which those roots may lie.

In fact the values of the roots themselves become rather irrelevant, and the necessary relations between coefficients and symmetric polynomial expressions can be found by computations in terms of symmetric polynomials only. An example of such relations are Newton's identitieswhich express the sum of any fixed power of the roots in terms of the elementary symmetric polynomials.

There are a few types of symmetric polynomials in the variables X 1X 2…, X n that are fundamental. For each nonnegative integer kthe elementary symmetric polynomial e k X 1…, X n is the sum of all distinct products of k distinct variables.Delve into our printable multiplying monomials worksheets for a wealth of practice in finding the product of any two monomials, a monomial by a binomial, and a monomial by a polynomial. Monomials, by the way, are algebraic expressions consisting of a single term.

They can be univariate or multivariate expressions. Abounding in exercises on single-variable and multivariable expressions, these pdfs help high school students perform monomial multiplication with ease. Grab some worksheets for free! Multiplying Monomials - Single Variable. In these multiplication of monomial by monomial pdf worksheets with 2 or 3 monomials of single variables, multiply the coefficients of one term with the other and add the exponents with the same base.

Multiplying Monomials - Multivariable. Multiplying multivariable monomials is best perfected with scads of practice! Regroup the factors - the coefficients, up to three variables and constants, and find the product of the monomials here.

Multiply Monomials by Binomials - Single Variable. Let your monomial multiplication skills be second to none! Multiply monomials with binomials - expressions with two terms. Multiply the monomial with each term inside the parentheses to obtain the product. Multiply Monomials by Binomials - Multivariable.

This resource thrives on multiplication of multivariate monomials and binomial expressions. Bring the product rule of exponents into play, perform the operation, and determine the product. Multiplying Monomials by Polynomials - Single Variable. Celebrate the diversely gifted monomials with these pdf worksheets presenting univariate monomials and polynomials! Distribute the monomial to every term of the polynomial and multiply them. Multiplying Monomials by Polynomials - Multivariable. High school students apply the distributive property and determine the product of the multivariable monomial and polynomial expressions that our printable multiplying monomials worksheets are chock-full of. Let your skills shine with a challenge packed in this set of multiplication equations consisting of missing monomials.

### Polynomials

Complete the multiplication statement by plugging in the missing factor or product. Members have exclusive facilities to download an individual worksheet, or an entire level. Login Become a Member. Multiplying Monomials - Single Variable In these multiplication of monomial by monomial pdf worksheets with 2 or 3 monomials of single variables, multiply the coefficients of one term with the other and add the exponents with the same base. Multiplying Monomials - Multivariable Multiplying multivariable monomials is best perfected with scads of practice!

Multiply Monomials by Binomials - Single Variable Let your monomial multiplication skills be second to none! Multiply Monomials by Binomials - Multivariable This resource thrives on multiplication of multivariate monomials and binomial expressions.

Multiplication Of Monomial By Polynomial

Multiplying Monomials by Polynomials - Single Variable Celebrate the diversely gifted monomials with these pdf worksheets presenting univariate monomials and polynomials! Multiplying Monomials by Polynomials - Multivariable High school students apply the distributive property and determine the product of the multivariable monomial and polynomial expressions that our printable multiplying monomials worksheets are chock-full of. Missing Monomials Let your skills shine with a challenge packed in this set of multiplication equations consisting of missing monomials.

What's New? Follow us. Not a Member?Permutation polynomials over finite fields constitute an active research area and have applications in many areas of science and engineering. In this paper, four classes of monomial complete permutation polynomials and one class of trinomial complete permutation polynomials are presented, one of which confirms a conjecture proposed by Wu et al.

Sci China Math —, Furthermore, we give two classes of permutation trinomial, and make some progress on a conjecture about the differential uniformity of power permutation polynomials proposed by Blondeau et al. Int J Inf Coding Theory —, This is a preview of subscription content, log in to check access.

Rent this article via DeepDyve. Berlekamp E. Control 10— Blondeau C. Coding Theory 1— Charpin P. SIAM J. Discret Math. Ding C.

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Theory Ser. A— China Ser. A 52— Akbary A. Finite Fields Appl.

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Dobbertin H. IEEE Trans. Theory 45— Kluwer, Dordrecht Fernando N. Helleseth T. Hollmann H. Hou X. Laigle-Chapuy Y. Lidl R. Encyclopedia of Mathematics and Its Applications, vol.

Niederreiter H. A 33— Theory 59— Sun J. Theory 51— China Math. Yuan J.This is the date and time in which the library was created with microsecond precision. A description of the status of the library. Example: 1 description optional A description of the script up to 8192 characters long. Example: "This is a description of my new script" imports optional A list of valid library identifiers. Example: "my new script" outputs optional A list of variables with their name, type, and optional description, defined in the source code of script, that will conform the outputs of execution.

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