Polynomial - Extensions of The Concept of A Polynomial

Extensions of The Concept of A Polynomial

Polynomials can involve more than one variable, in which they are called multivariate. Rings of polynomials in a finite number of variables are of fundamental importance in algebraic geometry which studies the simultaneous zero sets of several such multivariate polynomials. These rings can alternatively be constructed by repeating the construction of univariate polynomials with as coefficient ring another ring of polynomials: thus the ring R of polynomials in X and Y can be viewed as the ring (R) of polynomials in Y with as coefficients polynomials in X, or as the ring (R) of polynomials in X with as coefficients polynomials in Y. These identifications are compatible with arithmetic operations (they are isomorphisms of rings), but some notions such as degree or whether a polynomial is considered monic can change between these points of view. One can construct rings of polynomials in infinitely many variables, but since polynomials are (finite) expressions, any individual polynomial can only contain finitely many variables.

A binary polynomial where the second variable takes the form of an exponential function applied to the first variable, for example P(X,eX ), may be called an exponential polynomial.

Laurent polynomials are like polynomials, but allow negative powers of the variable(s) to occur.

Quotients of polynomials are called rational expressions (or rational fractions), and functions that evaluate rational expressions are called rational functions. Rational fractions are formal quotients of polynomials (they are formed from polynomials just as rational numbers are formed from integers, writing a fraction of two of them; fractions related by the canceling of common factors are identified with each other). The rational function defined by a rational fraction is the quotient of the polynomial functions defined by the numerator and the denominator of the rational fraction. The rational fractions contain the Laurent polynomials, but do not limit denominators to powers of a variable. While polynomial functions are defined for all values of the variables, a rational function is defined only for the values of the variables for which the denominator is not null. A rational function produces rational output for any rational input for which it is defined; this is not true of other functions such as trigonometric functions, logarithms and exponential functions.

Formal power series are like polynomials, but allow infinitely many non-zero terms to occur, so that they do not have finite degree. Unlike polynomials they cannot in general be explicitly and fully written down (just like real numbers cannot), but the rules for manipulating their terms are the same as for polynomials.

Read more about this topic:  Polynomial

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