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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.
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 ...
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 ...
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.
Beurling–Lax theorem (Hardy spaces) Bézout's theorem (algebraic geometry) Bing metrization theorem (general topology) Bing's recognition theorem (geometric topology) Binomial inverse theorem (linear algebra) Binomial theorem (algebra, combinatorics) Birch's theorem (algebraic number theory) Birkhoff–Grothendieck theorem (complex geometry)
Bézout's identity is a GCD related theorem, initially proved for the integers, which is valid for every principal ideal domain. In the case of the univariate polynomials over a field, it may be stated as follows.
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 .
Download as PDF; Printable version ... He was the first to state Bézout's theorem, ... and detailed an experiment in 1675 that showed when one side of a glass sheet ...