19 Facts About Multivariate polynomial

In mathematics, a polynomial is an expression consisting of indeterminates and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables.

Formally, the name of the Multivariate polynomial is P, not P, but the use of the functional notation P dates from a time when the distinction between a Multivariate polynomial and the associated function was unclear.

For example, "let P be a Multivariate polynomial" is a shorthand for "let P be a Multivariate polynomial in the indeterminate x".

Which is the polynomial function associated to P Frequently, when using this notation, one supposes that a is a number.

Exponent on an indeterminate in a term is called the degree of that indeterminate in that term; the degree of the term is the sum of the degrees of the indeterminates in that term, and the degree of a Multivariate polynomial is the largest degree of any term with nonzero coefficient.

Rather, the degree of the zero Multivariate polynomial is either left explicitly undefined, or defined as negative.

The zero Multivariate polynomial is unique in that it is the only Multivariate polynomial in one indeterminate that has an infinite number of roots.

The zero Multivariate polynomial is homogeneous, and, as a homogeneous Multivariate polynomial, its degree is undefined.

However, a real Multivariate polynomial function is a function from the reals to the reals that is defined by a real Multivariate polynomial.

Generally, unless otherwise specified, Multivariate polynomial functions have complex coefficients, arguments, and values.

In particular, a Multivariate polynomial, restricted to have real coefficients, defines a function from the complex numbers to the complex numbers.

Non-constant Multivariate polynomial function tends to infinity when the variable increases indefinitely.

Root of a nonzero univariate Multivariate polynomial is a value of such that = 0.

The number of roots of a nonzero Multivariate polynomial, counted with their respective multiplicities, cannot exceed the degree of, and equals this degree if all complex roots are considered.

Polynomials with more than one indeterminate, the combinations of values for the variables for which the Multivariate polynomial function takes the value zero are generally called zeros instead of "roots".

Trigonometric Multivariate polynomial is a finite linear combination of functions sin and cos with n taking on the values of one or more natural numbers.

Matrix Multivariate polynomial is a Multivariate polynomial with square matrices as variables.

Simple structure of Multivariate polynomial functions makes them quite useful in analyzing general functions using Multivariate polynomial approximations.

For example, in computational complexity theory the phrase Multivariate polynomial time means that the time it takes to complete an algorithm is bounded by a Multivariate polynomial function of some variable, such as the size of the input.