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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.
Coefficient: An expression multiplying one of the monomials of the polynomial. Root (or zero) of a polynomial: Given a polynomial p(x), the x values that satisfy p(x) = 0 are called roots (or zeroes) of the polynomial p. Graphing. End behaviour – Concavity – Orientation – Tangency point – Inflection point – Point where concavity changes.
One can obtain explicit formulas for the above expressions in the form of determinants, by considering the first n of Newton's identities (or it counterparts for the complete homogeneous polynomials) as linear equations in which the elementary symmetric functions are known and the power sums are unknowns (or vice versa), and apply Cramer's rule ...
The degree of the zero polynomial 0 (which has no terms at all) is generally treated as not defined (but see below). [9] For example: is a term. The coefficient is −5, the indeterminates are x and y, the degree of x is two, while the degree of y is one.
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The polynomial identities and invariants of n×n matrices. Regional Conference Series in Mathematics. Vol. 78. Providence, RI: American Mathematical Society. ISBN 0-8218-0730-7. Zbl 0714.16001. Kanel-Belov, Alexei; Rowen, Louis Halle (2005). Computational aspects of polynomial identities. Research Notes in Mathematics. Vol. 9. Wellesley, MA: A ...
where the above convention for the coefficients of the polynomials agrees with the definition of the binomial coefficients, because both give zero for all i > m and j > n, respectively. By comparing coefficients of x r, Vandermonde's identity follows for all integers r with 0 ≤ r ≤ m + n.
Here the greatest common divisor of 0 and 0 is taken to be 0.The integers x and y are called Bézout coefficients for (a, b); they are not unique.A pair of Bézout coefficients can be computed by the extended Euclidean algorithm, and this pair is, in the case of integers one of the two pairs such that | x | ≤ | b/d | and | y | ≤ | a/d |; equality occurs only if one of a and b is a multiple ...