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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 ...
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 polynomials in two or more variables, the degree of a term is the sum of the exponents of the variables in the term; the degree (sometimes called the total degree) of the polynomial is again the maximum of the degrees of all terms in the polynomial. For example, the polynomial x 2 y 2 + 3x 3 + 4y has degree 4, the same degree as the term x ...
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
In mathematics and computer science, Horner's method (or Horner's scheme) is an algorithm for polynomial evaluation.Although named after William George Horner, this method is much older, as it has been attributed to Joseph-Louis Lagrange by Horner himself, and can be traced back many hundreds of years to Chinese and Persian mathematicians. [1]
[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