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In algebra, a multilinear polynomial [1] is a multivariate polynomial that is linear (meaning affine) in each of its variables separately, but not necessarily simultaneously. It is a polynomial in which no variable occurs to a power of 2 {\displaystyle 2} or higher; that is, each monomial is a constant times a product of distinct variables.
In mathematics, polynomial identity testing (PIT) is the problem of efficiently determining whether two multivariate polynomials are identical. More formally, a PIT algorithm is given an arithmetic circuit that computes a polynomial p in a field , and decides whether p is the zero polynomial.
Moreover, the functional notation is often useful for specifying, in a single phrase, a polynomial and its indeterminate. For example, "let P(x) be a polynomial" is a shorthand for "let P be a polynomial in the indeterminate x". On the other hand, when it is not necessary to emphasize the name of the indeterminate, many formulas are much ...
Horner's method evaluates a polynomial using repeated bracketing: + + + + + = + (+ (+ (+ + (+)))). This method reduces the number of multiplications and additions to just Horner's method is so common that a computer instruction "multiply–accumulate operation" has been added to many computer processors, which allow doing the addition and multiplication operations in one combined step.
Conversely, shows that any integer-valued polynomial is an integer linear combination of these binomial coefficient polynomials. More generally, for any subring R of a characteristic 0 field K , a polynomial in K [ t ] takes values in R at all integers if and only if it is an R -linear combination of binomial coefficient polynomials.
Such a chain consists of a sequence of polynomials f 1 (x 1), f 2 (x 1, x 2), ..., f n (x 1, ..., x n) such that, for every i such that 1 ≤ i ≤ n. f i is a polynomial in x 1, ..., x i only, which has a degree d i > 0 in x i; the coefficient of x i d i in f i is a polynomial in x 1, ..., x i −1 which does not have any common zero with f 1 ...
Polynomial interpolation also forms the basis for algorithms in numerical quadrature (Simpson's rule) and numerical ordinary differential equations (multigrid methods). In computer graphics, polynomials can be used to approximate complicated plane curves given a few specified points, for example the shapes of letters in typography.
A typical example of a Chebyshev space is the subspace of Chebyshev polynomials of order n in the space of real continuous functions on an interval, C[a, b]. The polynomial of best approximation within a given subspace is defined to be the one that minimizes the maximum absolute difference between the polynomial and the function.
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