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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.
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 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 ...
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 ...
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.
Except for Bézout's theorem, the general approach was to eliminate variables for reducing the problem to a single equation in one variable. The case of linear equations was completely solved by Gaussian elimination , where the older method of Cramer's rule does not proceed by elimination, and works only when the number of equations equals the ...
It was this very translation in which Fermat wrote his famous margin note claiming that he had a proof of Fermat's Last Theorem. The same text renders Diophantus' term παρισὀτης as adaequalitat , which became Fermat's technique of adequality , a pioneering method of infinitesimal calculus .
Theorem - If a projective hypersurface of degree d does not contain any irreducible component of an algebraic set of degree δ, then the degree of their intersection is dδ. The usual Bézout's theorem is easily deduced by starting from a hypersurface, and intersecting it with n − 1 other hypersurfaces, one after the other.