Ring theory | Polynomial functions

Ring of polynomial functions

In mathematics, the ring of polynomial functions on a vector space V over a field k gives a coordinate-free analog of a polynomial ring. It is denoted by k[V]. If V is finite dimensional and is viewed as an algebraic variety, then k[V] is precisely the coordinate ring of V. The explicit definition of the ring can be given as follows. If is a polynomial ring, then we can view as coordinate functions on ; i.e., when This suggests the following: given a vector space V, let k[V] be the commutative k-algebra generated by the dual space , which is a subring of the ring of all functions . If we fix a basis for V and write for its dual basis, then k[V] consists of polynomials in . If k is infinite, then k[V] is the symmetric algebra of the dual space . In applications, one also defines k[V] when V is defined over some subfield of k (e.g., k is the complex field and V is a real vector space.) The same definition still applies. Throughout the article, for simplicity, the base field k is assumed to be infinite. (Wikipedia).

Video thumbnail

Determine if a Function is a Polynomial Function

This video explains how to determine if a function is a polynomial function. http://mathispower4u.com

From playlist Determining the Characteristics of Polynomial Functions

Video thumbnail

Determine the Domain of Various Functions

This video explains how to determine the domain of various functions including linear, quadratic, square root, cube root, cubic, absolute value, and rational functions. http://mathispower4u.com

From playlist The Properties of Functions

Video thumbnail

Classify a polynomial then determining if it is a polynomial or not

πŸ‘‰ Learn how to determine whether a given equation is a polynomial or not. A polynomial function or equation is the sum of one or more terms where each term is either a number, or a number times the independent variable raised to a positive integer exponent. A polynomial equation of functio

From playlist Is it a polynomial or not?

Video thumbnail

Rational Functions

In this video we cover some rational function fundamentals, including asymptotes and interecepts.

From playlist Polynomial Functions

Video thumbnail

Determining if a function is a polynomial or not then determine degree and LC

πŸ‘‰ Learn how to determine whether a given equation is a polynomial or not. A polynomial function or equation is the sum of one or more terms where each term is either a number, or a number times the independent variable raised to a positive integer exponent. A polynomial equation of functio

From playlist Is it a polynomial or not?

Video thumbnail

Determining if a equation is a polynomial or not

πŸ‘‰ Learn how to determine whether a given equation is a polynomial or not. A polynomial function or equation is the sum of one or more terms where each term is either a number, or a number times the independent variable raised to a positive integer exponent. A polynomial equation of functio

From playlist Is it a polynomial or not?

Video thumbnail

Is it a polynomial with two variables

πŸ‘‰ Learn how to determine whether a given equation is a polynomial or not. A polynomial function or equation is the sum of one or more terms where each term is either a number, or a number times the independent variable raised to a positive integer exponent. A polynomial equation of functio

From playlist Is it a polynomial or not?

Video thumbnail

Dividing Polynomials Part 2

In this video we look at some formal definitions of polynomial division.

From playlist Polynomial Functions

Video thumbnail

Learn how to identify if a function is a polynomial and identify the degree and LC

πŸ‘‰ Learn how to determine whether a given equation is a polynomial or not. A polynomial function or equation is the sum of one or more terms where each term is either a number, or a number times the independent variable raised to a positive integer exponent. A polynomial equation of functio

From playlist Is it a polynomial or not?

Video thumbnail

Rings 17 Noetherian rings

This lecture is part of an online course on rings and modules. We define Noetherian rings, give several equivalent properties, and give some examples of rings that are or are not Noetherian. This will be continued in the next lecture about Hilbert's finiteness theorems. For the other

From playlist Rings and modules

Video thumbnail

Duality in Algebraic Geometry by Suresh Nayak

PROGRAM DUALITIES IN TOPOLOGY AND ALGEBRA (ONLINE) ORGANIZERS: Samik Basu (ISI Kolkata, India), Anita Naolekar (ISI Bangalore, India) and Rekha Santhanam (IIT Mumbai, India) DATE & TIME: 01 February 2021 to 13 February 2021 VENUE: Online Duality phenomena are ubiquitous in mathematics

From playlist Dualities in Topology and Algebra (Online)

Video thumbnail

Commutative algebra 5 (Noetherian rings)

This lecture is part of an online course on commutative algebra, following the book "Commutative algebra with a view toward algebraic geometry" by David Eisenbud. In this lecture we find three equivalent ways of defining Noetherian rings, and give several examples of Noetherian and non-No

From playlist Commutative algebra

Video thumbnail

Nonlinear algebra, Lecture 10: "Invariant Theory", by Bernd Sturmfels

This is the tenth lecture in the IMPRS Ringvorlesung, the advanced graduate course at the Max Planck Institute for Mathematics in the Sciences.

From playlist IMPRS Ringvorlesung - Introduction to Nonlinear Algebra

Video thumbnail

Rings and modules 2: Group rings

This lecture is part of an online course on rings and modules. We decribe some examples of rings constructed from groups and monoids, such as group rings and rings of Dirichlet polynomials. For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EFj52XDLrm

From playlist Rings and modules

Video thumbnail

Herwig Hauser : Commutative algebra for Artin approximation - Part 2

Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b

From playlist Jean-Morlet Chair - Hauser/Rond

Video thumbnail

Nonlinear algebra, Lecture 2: "Algebraic Varieties", by Mateusz MichaΕ‚ek

This is the second lecture in the IMPRS Ringvorlesung, the advanced graduate course at the Max Planck Institute for Mathematics in the Sciences. In this lecture, Mateusz Michalek describes the main characters in algebraic geometry: algebraic varieties.

From playlist IMPRS Ringvorlesung - Introduction to Nonlinear Algebra

Video thumbnail

Markus Rosenkranz Talk 1 7/7/14 Part 3

Title: Integro-Differential Polynomials and Free Integro-Differential Algebras

From playlist Spring 2014

Video thumbnail

"New Paradigms in Invariant Theory" - Roger Howe, Yale University [2011]

HKUST Institute for Advanced Study Distinguished Lecture New Paradigms in Invariant Theory Speaker: Prof Roger Howe, Yale University Date: 13/6/2011 Video taken from: http://video.ust.hk/Watch.aspx?Video=6A41D5F6B1A790DC

From playlist Mathematics

Video thumbnail

Polynomial Functions: Domain and Range

How to find the domain and range for any polynomial function with a simple flowchart.

From playlist Domain and Range of Functions

Related pages

Operator product expansion | Vector space | Subring | Associative algebra | Zariski tangent space | Algebraic variety | Homogeneous polynomial | Algebra homomorphism | Isomorphism | Radius of convergence | Polynomial | Partial derivative | Algebraic geometry of projective spaces | Polynomial ring | Degree of a polynomial | General linear group | Field extension | Linear map | Dimension (vector space) | Characteristic (algebra) | Mathematics | Function (mathematics) | Dual space | Field (mathematics) | Set (mathematics) | Real number | Section (fiber bundle) | Ring (mathematics) | Taylor series | Vector bundle | Basis (linear algebra) | Quadratic form | Complex number | Symmetric algebra | Endomorphism