**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Concept# Monomial basis

Summary

In mathematics the monomial basis of a polynomial ring is its basis (as a vector space or free module over the field or ring of coefficients) that consists of all monomials. The monomials form a basis because every polynomial may be uniquely written as a finite linear combination of monomials (this is an immediate consequence of the definition of a polynomial).
The polynomial ring K[x] of univariate polynomials over a field K is a K-vector space, which has
as an (infinite) basis. More generally, if K is a ring then K[x] is a free module which has the same basis.
The polynomials of degree at most d form also a vector space (or a free module in the case of a ring of coefficients), which has as a basis.
The canonical form of a polynomial is its expression on this basis:
or, using the shorter sigma notation:
The monomial basis is naturally totally ordered, either by increasing degrees
or by decreasing degrees
In the case of several indeterminates a monomial is a product
where the are non-negative integers. As an exponent equal to zero means that the corresponding indeterminate does not appear in the monomial; in particular is a monomial.
Similar to the case of univariate polynomials, the polynomials in form a vector space (if the coefficients belong to a field) or a free module (if the coefficients belong to a ring), which has the set of all monomials as a basis, called the monomial basis.
The homogeneous polynomials of degree form a subspace which has the monomials of degree as a basis. The dimension of this subspace is the number of monomials of degree , which is
where is a binomial coefficient.
The polynomials of degree at most form also a subspace, which has the monomials of degree at most as a basis. The number of these monomials is the dimension of this subspace, equal to
In contrast to the univariate case, there is no natural total order of the monomial basis in the multivariate case. For problems which require choosing a total order, such as Gröbner basis computations, one generally chooses an admissible mon

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related courses (5)

Related concepts (9)

Related lectures (53)

MATH-111(e): Linear Algebra

L'objectif du cours est d'introduire les notions de base de l'algèbre linéaire et ses applications.

MATH-410: Riemann surfaces

This course is an introduction to the theory of Riemann surfaces. Riemann surfaces naturally appear is mathematics in many different ways: as a result of analytic continuation, as quotients of complex

MATH-494: Topics in arithmetic geometry

P-adic numbers are a number theoretic analogue of the real numbers, which interpolate between arithmetics, analysis and geometry. In this course we study their basic properties and give various applic

In numerical analysis, the Lagrange interpolating polynomial is the unique polynomial of lowest degree that interpolates a given set of data. Given a data set of coordinate pairs with the are called nodes and the are called values. The Lagrange polynomial has degree and assumes each value at the corresponding node, Although named after Joseph-Louis Lagrange, who published it in 1795, the method was first discovered in 1779 by Edward Waring. It is also an easy consequence of a formula published in 1783 by Leonhard Euler.

In the mathematical field of numerical analysis, a Bernstein polynomial is a polynomial that is a linear combination of Bernstein basis polynomials. The idea is named after Sergei Natanovich Bernstein. A numerically stable way to evaluate polynomials in Bernstein form is de Casteljau's algorithm. Polynomials in Bernstein form were first used by Bernstein in a constructive proof for the Weierstrass approximation theorem. With the advent of computer graphics, Bernstein polynomials, restricted to the interval [0, 1], became important in the form of Bézier curves.

In the mathematical field of numerical analysis, a Newton polynomial, named after its inventor Isaac Newton, is an interpolation polynomial for a given set of data points. The Newton polynomial is sometimes called Newton's divided differences interpolation polynomial because the coefficients of the polynomial are calculated using Newton's divided differences method. Given a set of k + 1 data points where no two xj are the same, the Newton interpolation polynomial is a linear combination of Newton basis polynomials with the Newton basis polynomials defined as for j > 0 and .

Harmonic Forms: Main Theorem

Explores harmonic forms on Riemann surfaces and the uniqueness of solutions to harmonic equations.

Extension of Domains

Covers extending domains, leading coefficients, and valid extensions.

Polynomial Optimization: SOS and Nonnegative Polynomials

Explores polynomial optimization, emphasizing SOS and nonnegative polynomials, including the representation of polynomials as quadratic functions of monomials.