Search results
Results from the WOW.Com Content Network
Bézout's theorem is a statement in algebraic geometry concerning the number of common zeros of n polynomials in n indeterminates. In its original form the theorem states that in general the number of common zeros equals the product of the degrees of the polynomials. [1] It is named after Étienne Bézout.
Here the greatest common divisor of 0 and 0 is taken to be 0.The integers x and y are called Bézout coefficients for (a, b); they are not unique.A pair of Bézout coefficients can be computed by the extended Euclidean algorithm, and this pair is, in the case of integers one of the two pairs such that | x | ≤ | b/d | and | y | ≤ | a/d |; equality occurs only if one of a and b is a multiple ...
A second difference lies in the bound on the size of the Bézout coefficients provided by the extended Euclidean algorithm, which is more accurate in the polynomial case, leading to the following theorem. If a and b are two nonzero polynomials, then the extended Euclidean algorithm produces the unique pair of polynomials (s, t) such that
A generalization of Bézout's theorem asserts that, if an intersection of n projective hypersurfaces has codimension n, then the degree of the intersection is the product of the degrees of the hypersurfaces. The degree of a projective variety is the evaluation at 1 of the numerator of the Hilbert series of its coordinate ring.
Setting = + +, the multi-homogeneous Bézout theorem is the following. With above notation, n multi-homogeneous polynomials of multi-degrees d 1 , … , d n {\displaystyle \mathbf {d} _{1},\ldots ,\mathbf {d} _{n}} define either a multi-projective algebraic set of positive dimension, or a zero-dimensional algebraic set consisting of B points ...
In algebra, the polynomial remainder theorem or little Bézout's theorem (named after Étienne Bézout) [1] is an application of Euclidean division of polynomials.It states that, for every number , any polynomial is the sum of () and the product of and a polynomial in of degree one less than the degree of .
Here we are using Hilbert series of filtered algebras, and the fact that the Hilbert series of a graded algebra is also its Hilbert series as filtered algebra. Thus R 0 {\displaystyle R_{0}} is an Artinian ring , which is a k -vector space of dimension P (1) , and Jordan–Hölder theorem may be used for proving that P (1) is the degree of the ...
Diophantine geometry, is the application of techniques from algebraic geometry which considers equations that also have a geometric meaning. The central idea of Diophantine geometry is that of a rational point , namely a solution to a polynomial equation or a system of polynomial equations , which is a vector in a prescribed field K , when K is ...