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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.
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems. Classically, it studies zeros of multivariate polynomials ; the modern approach generalizes this in a few different aspects.
Wiles's proof uses many techniques from algebraic geometry and number theory and has many ramifications in these branches of mathematics. It also uses standard constructions of modern algebraic geometry such as the category of schemes , significant number theoretic ideas from Iwasawa theory , and other 20th-century techniques which were not ...
In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros", or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic geometry. It relates algebraic sets to ideals in polynomial rings over algebraically closed fields.
Fermat's little theorem and some proofs; Gödel's completeness theorem and its original proof; Mathematical induction and a proof; Proof that 0.999... equals 1; Proof that 22/7 exceeds π; Proof that e is irrational; Proof that π is irrational; Proof that the sum of the reciprocals of the primes diverges
In the special case when k is the function field of an algebraic curve over a finite field and f is any character that is trivial on k, this recovers the geometric Riemann–Roch theorem. [ 12 ] Other versions of the arithmetic Riemann–Roch theorem make use of Arakelov theory to resemble the traditional Riemann–Roch theorem more exactly.
The theorem is interpreted in algebraic geometry as follows: every algebraic set is the set of the common zeros of finitely many polynomials. Hilbert's proof is highly non-constructive : it proceeds by induction on the number of variables, and, at each induction step uses the non-constructive proof for one variable less.
Castelnuovo theorem (algebraic geometry) Castelnuovo–de Franchis theorem (algebraic geometry) Castigliano's first and second theorems (structural analysis) Cauchy integral theorem (complex analysis) Cauchy–Hadamard theorem (complex analysis) Cauchy–Kowalevski theorem (partial differential equations) Cauchy's theorem