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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.
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 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]
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 ...
[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 case of the 105th cyclotomic polynomial is interesting because 105 is the least positive integer that is the product of three distinct odd prime numbers (3×5×7) and this polynomial is the first one that has a coefficient other than 1, 0, or −1: [3]
So we have a linear map which agrees with the inverse of on the column space; hence it agrees with on the span of the column space. Since is invertible, the column vectors span all of , so our map really is the inverse of . Cramer's rule follows.
The fundamental theorem of symmetric polynomials implies that the are algebraic independent, and thus that the map that sends each to the corresponding is a field isomorphism from F to K. This means that one may consider P ( x ) = 0 {\displaystyle P(x)=0} as a generic equation.