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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.
Many other theorems in elementary number theory, such as Euclid's lemma or the Chinese remainder theorem, result from Bézout's identity. A Bézout domain is an integral domain in which Bézout's identity holds. In particular, Bézout's identity holds in principal ideal domains. Every theorem that results from Bézout's identity is thus true in ...
For stating the generalization of Bézout's theorem, it is convenient to introduce new indeterminates , …,, and to represent the multi-degree , …, by the linear form = + +. In the following, "multi-degree" will refer to this linear form rather than to the sequence of degrees.
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 .
Upload file; Special pages; Permanent link; Page information; Get shortened URL; Download QR code; Print/export Download as PDF; ... Multi-homogeneous Bézout theorem;
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 ...
Fermat's last theorem Fermat's last theorem, one of the most famous and difficult to prove theorems in number theory, states that for any integer n > 2, the equation a n + b n = c n has no positive integer solutions. Fermat's little theorem Fermat's little theorem field extension A field extension L/K is a pair of fields K and L such that K is ...