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The above formula defines the derivative of a polynomial even if the coefficients belong to a ... The skew-polynomial ring is defined similarly for a ring R and a ...
Informally, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.
A matrix polynomial equation is an equality between two matrix polynomials, which holds for the specific matrices in question. A matrix polynomial identity is a matrix polynomial equation which holds for all matrices A in a specified matrix ring M n (R).
When the ring R of scalars is commutative, there is an alternative and equivalent definition of the formal derivative, which resembles the one seen in differential calculus. The element Y–X of the ring R[X,Y] divides Y n – X n for any nonnegative integer n, and therefore divides f(Y) – f(X) for any polynomial f in one
In mathematics, the ring of polynomial functions on a vector space V over a field k gives a coordinate-free analog of a polynomial ring. It is denoted by k[V]. If V is finite dimensional and is viewed as an algebraic variety, then k[V] is precisely the coordinate ring of V. The explicit definition of the ring can be given as follows.
The fact that is an irreducible polynomial of degree () in the ring [] is a nontrivial result due to Gauss. [4] Depending on the chosen definition, it is either the value of the degree or the irreducibility which is a nontrivial result.
(A polynomial with integer coefficients is primitive if it has 1 as a greatest common divisor of its coefficients. [note 2]) A corollary of Gauss's lemma, sometimes also called Gauss's lemma, is that a primitive polynomial is irreducible over the integers if and only if it is irreducible over the rational numbers. More generally, a primitive ...
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 A quasi-projective variety is a Zariski open subset of a projective variety.