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When V, W are finite-dimensional vector spaces and are viewed as algebraic varieties, then a polynomial mapping is precisely a morphism of algebraic varieties. One fundamental outstanding question regarding polynomial mappings is the Jacobian conjecture, which concerns the sufficiency of a polynomial mapping to be invertible.
The word polynomial joins two diverse roots: the Greek poly, meaning "many", and the Latin nomen, or "name". It was derived from the term binomial by replacing the Latin root bi-with the Greek poly-. That is, it means a sum of many terms (many monomials). The word polynomial was first used in the 17th century. [6]
For univariate polynomials over the rationals (or more generally over a field of characteristic zero), Yun's algorithm exploits this to efficiently factorize the polynomial into square-free factors, that is, factors that are not a multiple of a square, performing a sequence of GCD computations starting with gcd(f(x), f '(x)). To factorize the ...
The strong real Jacobian conjecture was that a real polynomial map with a nowhere vanishing Jacobian determinant has a smooth global inverse. That is equivalent to asking whether such a map is topologically a proper map, in which case it is a covering map of a simply connected manifold, hence invertible. Sergey Pinchuk constructed two variable ...
That is, a regular map : is the same as the restriction of a polynomial map whose components satisfy the defining equations of . More generally, a map f : X → Y between two varieties is regular at a point x if there is a neighbourhood U of x and a neighbourhood V of f ( x ) such that f ( U ) ⊂ V and the restricted function f : U → V is ...
The most widely used orthogonal polynomials are the classical orthogonal polynomials, consisting of the Hermite polynomials, the Laguerre polynomials and the Jacobi polynomials. The Gegenbauer polynomials form the most important class of Jacobi polynomials; they include the Chebyshev polynomials, and the Legendre polynomials as special
[1]: 10 Projective varieties are also equipped with the Zariski topology by declaring all algebraic sets to be closed. Given a subset V of P n, let I(V) be the ideal generated by all homogeneous polynomials vanishing on V. For any projective algebraic set V, the coordinate ring of V is the quotient of the polynomial ring by this ideal. [1]: 10
The pair (V, Q) consisting of a finite-dimensional vector space V over K and a quadratic map Q from V to K is called a quadratic space, and B as defined here is the associated symmetric bilinear form of Q. The notion of a quadratic space is a coordinate-free version of the notion of quadratic form. Sometimes, Q is also called a quadratic form.