mathematics, the Ax–Grothendieck theorem is a result about injectivity and surjectivity of polynomials that was proved independently by James Ax and Alexander...

{\displaystyle p.} One consequence is the following special case of the Ax–Grothendieck theorem: all injective complex polynomials C n → C n {\displaystyle \mathbb...

James Burton Ax (10 January 1937 – 11 June 2006) was an American mathematician who made groundbreaking contributions in algebra and number theory using...

Alexander Grothendieck (1928–2014) is the eponym of many things. Ax–Grothendieck theorem Birkhoff–Grothendieck theorem Brieskorn–Grothendieck resolution...

other applied fields. Ax–Grothendieck theorem (model theory) Barwise compactness theorem (mathematical logic) Borel determinacy theorem (set theory)...

logic) Differentially closed field Exponential field Ax–Grothendieck theorem Ax–Kochen theorem Peano axioms Non-standard model of arithmetic First-order...

arguments such as vanishing theorems must be used. Arakelov theory Grothendieck–Riemann–Roch theorem Hirzebruch–Riemann–Roch theorem Kawasaki's Riemann–Roch...

Tarski–Grothendieck set theory (TG, named after mathematicians Alfred Tarski and Alexander Grothendieck) is an axiomatic set theory. It is a non-conservative...

non-surjective function (also not a bijection) Mathematics portal Ax–Grothendieck theorem Bijection, injection and surjection Bijective numeration Bijective...

In mathematics, the Grothendieck inequality states that there is a universal constant K G {\displaystyle K_{G}} with the following property. If Mij is...

introduced formally through the construction that was discovered by Grothendieck (see Grothendieck group). There is a natural homomorphism GW → Z given by dimension:...

uses it to analyze the concept of randomness. 1966 - Grothendieck proves the Ax-Grothendieck theorem: any injective polynomial self-map of algebraic varieties...

In mathematics, Richardson's theorem establishes the undecidability of the equality of real numbers defined by expressions involving integers, π, ln 2...

In functional analysis, the Hahn–Banach theorem is a central result that allows the extension of bounded linear functionals defined on a vector subspace...

Eden theorem asserts that every injective cellular automaton is surjective. It can be proven for sofic groups using the Ax–Grothendieck theorem, an analogous...

2014, it is still unknown whether every group is surjunctive. Ax–Grothendieck theorem, an analogous result for polynomials Ceccherini-Silberstein & Coornaert...

special case of Artin's conjecture on diophantine equations, the Ax–Kochen theorem. The ultraproduct construction also led to Abraham Robinson's development...

James Ax POLY Mathematics professor at Stanford University and Cornell University; won Frank Nelson Cole Prize in Number Theory; proved Ax–Grothendieck theorem...

is also true for almost all Fp((t)). An application of this is the Ax–Kochen theorem describing zeros of homogeneous polynomials in Qp. Tamely ramified...

For example, Wiles's proof of Fermat's Last Theorem implicitly relies on the existence of Grothendieck universes, very large infinite sets, for solving...

the Riemann–Roch theorem. In the 1960s, Grothendieck defined the notion of a site, meaning a category equipped with a Grothendieck topology. A site C...

variety, semistable elliptic curve, Serre–Tate theorem. Grothendieck–Katz conjecture The Grothendieck–Katz p-curvature conjecture applies reduction modulo...

(see Nouveaux Essais, IV,2)" (ibid p 421) The principle was stated as a theorem of propositional logic by Russell and Whitehead in Principia Mathematica...

gave proofs of versions of the open mapping theorem, closed graph theorem, and Hahn–Banach theorem. Grothendieck, Alexander (1955). "Produits Tensoriels Topologiques...

consequence of conjectural results in the theory of motives. In this setting Grothendieck's period conjecture for an abelian variety A states that the transcendence...

Riemann–Roch theorem to higher dimension is the Hirzebruch–Riemann–Roch theorem, as well as the far-reaching Grothendieck–Riemann–Roch theorem. Hilbert schemes...

at the International Congress of Mathematicians in Nice (The regularity theorem in algebraic geometry) and in 1978 in Helsinki (p-adic L functions, Serre-Tate...

program and Gödel's incompleteness theorem in the 1920s and 30s. First, note that Birkhoff proved with his HSP theorem that, remarkably, the equational...

finitely axiomatizable, while ZFC and MK are not. A key theorem of NBG is the class existence theorem, which states that for every formula whose quantifiers...

calculus correspond to relevant logic. The local truth (∇) modality in Grothendieck topology or the equivalent "lax" modality (◯) of Benton, Bierman, and...