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This is called Euclidean division, division with remainder or polynomial long division and shows that the ring F[x] is a Euclidean domain. Analogously, prime polynomials (more correctly, irreducible polynomials) can be defined as non-zero polynomials which cannot be factorized into the product of two non-constant polynomials.
(t maps to x) where ¯ is the polynomial function defined by f. The resulting map is injective if and only if R is infinite. Given a non-constant monic polynomial f in R[t], there exists a ring S containing R such that f is a product of linear factors in S[t]. [38] Let k be an algebraically closed field.
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 ...
As in calculus, the derivative detects multiple roots. If R is a field then R[x] is a Euclidean domain, and in this situation we can define multiplicity of roots; for every polynomial f(x) in R[x] and every element r of R, there exists a nonnegative integer m r and a polynomial g(x) such that = ()
In the special case of the zero polynomial, all of whose coefficients are zero, the leading coefficient is undefined, and the degree has been variously left undefined, [9] defined to be −1, [10] or defined to be a −∞. [11] A constant polynomial is either the zero polynomial, or a polynomial of degree zero.
For every φ: X → Y the map φ an defined above is a mapping of analytic spaces. Furthermore, the map φ ↦ φ an maps open immersions into open immersions. If X = Spec( C [ x 1 ,..., x n ]) then X an = C n and O X a n ( U ) {\displaystyle {\mathcal {O}}_{X}^{\mathrm {an} }(U)} for every polydisc U is a suitable quotient of the space of ...
The process of interpolation maps the function f to a polynomial p. This defines a mapping X from the space C([a, b]) of all continuous functions on [a, b] to itself. The map X is linear and it is a projection on the subspace () of polynomials of degree n or less. The Lebesgue constant L is defined as the operator norm of X.
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