Ax–Kochen theorem
Encyclopedia
The Ax–Kochen theorem, named for James Ax
and Simon B. Kochen
, states that for each positive integer d there is a finite set Yd of prime numbers, such that if p is any prime not in Yd then every homogeneous polynomial of degree d over the p-adic number
s in at least d2+1 variables has a nontrivial zero.
, such as model theory
.
One first proves Serge Lang
's theorem, stating that the analogous theorem is true for the field Fp((t)) of formal Laurent series
over a finite field
Fp with
. In other words, every homogeneous polynomial of degree d with more than d2 variables has a non-trivial zero (so Fp((t)) is a C2 field
).
Then one shows that if two Henselian
valued fields have equivalent valuation groups and residue fields, and the residue fields have characteristic
0, then they are equivalent (which means that a first order sentence is true for one if and only if it is true for the other).
Next one applies this to two fields, one given by an ultraproduct
over all primes of the fields Fp((t)) and the other given by an ultraproduct over all primes of the p-adic fields Qp.
Both residue fields are given by an ultraproduct over the fields Fp, so are isomorphic and have characteristic 0, and both value groups are the same, so the ultraproducts are equivalent. (Taking ultraproducts is used to force the residue field to have characteristic 0; the residue fields of Fp((t))
and Qp both have non-zero characteristic p.)
The equivalence of these ultraproducts implies that for any sentence in the language of valued fields, there is a finite set Y of exceptional primes, such that for any p not in this set the sentence is true for Fp((t)) if and only if it is true for the field of p-adic numbers. Applying this to the sentence stating that
every non-constant homogeneous polynomial of degree d in at least d2+1 variables represents 0, and using Lang's theorem,
one gets the Ax-Kochen theorem.
found a purely geometric proof for a conjecture of Jean-Louis Colliot-Thélène
which generalizes the Ax-Kochen theorem. He presented his proof at the "Variétés rationnelles" seminar at École Normale Supérieure in Paris, but the proof has not been published yet.
conjectured this theorem without the finite exceptional set Yd, but Guy Terjanian
found the following 2-adic counterexample for d = 4. Define
Then G has the property that it is 1 mod 4 if some x is odd, and 0 mod 16 otherwise.
It follows easily from this that the homogeneous form
of degree d=4 in 18> d2 variables has no non-trivial zeros over the 2-adic integers.
Later Terjanian showed that for each prime p and multiple d>2 of p(p−1), there is a form over the p-adic numbers of degree d with more than d2 variables but no nontrivial zeros. In other words, for all d> 2, Yd contains all primes p such that p(p−1) divides d.
gave an explicit but very large bound for the exceptional set of primes p. If the degree d is 1, 2, or 3 the exceptional set is empty. showed that if d=5 the exceptional set is bounded by 13, and showed that for d=7 the exceptional set is bounded by 883 and for d=11 it is bounded by 8053.
James Ax
James Burton Ax was a mathematician who proved several results in algebra and number theory by using model theory. He shared the seventh Frank Nelson Cole Prize in Number Theory with Simon B. Kochen, which was awarded for a series of three joint papers on Diophantine problems.James Ax earned his...
and Simon B. Kochen
Simon B. Kochen
Simon Bernhard Kochen is an American mathematician, working in the fields of model theory, number theory and quantum mechanics....
, states that for each positive integer d there is a finite set Yd of prime numbers, such that if p is any prime not in Yd then every homogeneous polynomial of degree d over the p-adic number
P-adic number
In mathematics, and chiefly number theory, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems...
s in at least d2+1 variables has a nontrivial zero.
The proof of the theorem
The proof of the theorem makes extensive use of methods from mathematical logicMathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
, such as model theory
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
.
One first proves Serge Lang
Serge Lang
Serge Lang was a French-born American mathematician. He was known for his work in number theory and for his mathematics textbooks, including the influential Algebra...
's theorem, stating that the analogous theorem is true for the field Fp((t)) of formal Laurent series
Laurent series
In mathematics, the Laurent series of a complex function f is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where...
over a finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
Fp with
. In other words, every homogeneous polynomial of degree d with more than d2 variables has a non-trivial zero (so Fp((t)) is a C2 fieldQuasi-algebraically closed field
In mathematics, a field F is called quasi-algebraically closed if every non-constant homogeneous polynomial P over F has a non-trivial zero provided the number of its variables is more than its degree....
).
Then one shows that if two Henselian
Hensel's lemma
In mathematics, Hensel's lemma, also known as Hensel's lifting lemma, named after Kurt Hensel, is a result in modular arithmetic, stating that if a polynomial equation has a simple root modulo a prime number , then this root corresponds to a unique root of the same equation modulo any higher power...
valued fields have equivalent valuation groups and residue fields, and the residue fields have characteristic
Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
0, then they are equivalent (which means that a first order sentence is true for one if and only if it is true for the other).
Next one applies this to two fields, one given by an ultraproduct
Ultraproduct
The ultraproduct is a mathematical construction that appears mainly in abstract algebra and in model theory, a branch of mathematical logic. An ultraproduct is a quotient of the direct product of a family of structures. All factors need to have the same signature...
over all primes of the fields Fp((t)) and the other given by an ultraproduct over all primes of the p-adic fields Qp.
Both residue fields are given by an ultraproduct over the fields Fp, so are isomorphic and have characteristic 0, and both value groups are the same, so the ultraproducts are equivalent. (Taking ultraproducts is used to force the residue field to have characteristic 0; the residue fields of Fp((t))
and Qp both have non-zero characteristic p.)
The equivalence of these ultraproducts implies that for any sentence in the language of valued fields, there is a finite set Y of exceptional primes, such that for any p not in this set the sentence is true for Fp((t)) if and only if it is true for the field of p-adic numbers. Applying this to the sentence stating that
every non-constant homogeneous polynomial of degree d in at least d2+1 variables represents 0, and using Lang's theorem,
one gets the Ax-Kochen theorem.
Alternative proof
In 2008, Jan DenefJan Denef
Jan Denef is a Belgian mathematician. He is Full Professor of Mathematics at the Katholieke Universiteit Leuven.He is a specialist of model theory, number theory and algebraic geometry. He is well known for his early work on Hilbert's tenth problemand for developing the theory of motivic...
found a purely geometric proof for a conjecture of Jean-Louis Colliot-Thélène
Jean-Louis Colliot-Thélène
Jean-Louis Colliot-Thélène is a French mathematician, born on 2 December 1947. He is a Directeur de Recherches at CNRS at the Université Paris-Sud in Orsay.He studies mainly number theory and algebraic geometry with an arithmetic flavor.-Awards:...
which generalizes the Ax-Kochen theorem. He presented his proof at the "Variétés rationnelles" seminar at École Normale Supérieure in Paris, but the proof has not been published yet.
Exceptional primes
Emil ArtinEmil Artin
Emil Artin was an Austrian-American mathematician of Armenian descent.-Parents:Emil Artin was born in Vienna to parents Emma Maria, née Laura , a soubrette on the operetta stages of Austria and Germany, and Emil Hadochadus Maria Artin, Austrian-born of Armenian descent...
conjectured this theorem without the finite exceptional set Yd, but Guy Terjanian
Guy Terjanian
Guy Terjanian is a French-Armenian mathematician who has worked on algebraic number theory. He achieved his Ph.D under Claude Chevalley in 1966, and at that time published a counterexample to the original form of a conjecture of Emil Artin, which suitably modified had just been proved as the...
found the following 2-adic counterexample for d = 4. Define
- G(x) = G(x1, x2, x3) =Σ xi4 − Σi<j xi2xj2 − x1x2x3(x1 + x2 + x3).
Then G has the property that it is 1 mod 4 if some x is odd, and 0 mod 16 otherwise.
It follows easily from this that the homogeneous form
- G(x) + G(y) + G(z) + 4G(u) + 4G(v) + 4G(w)
of degree d=4 in 18> d2 variables has no non-trivial zeros over the 2-adic integers.
Later Terjanian showed that for each prime p and multiple d>2 of p(p−1), there is a form over the p-adic numbers of degree d with more than d2 variables but no nontrivial zeros. In other words, for all d> 2, Yd contains all primes p such that p(p−1) divides d.
gave an explicit but very large bound for the exceptional set of primes p. If the degree d is 1, 2, or 3 the exceptional set is empty. showed that if d=5 the exceptional set is bounded by 13, and showed that for d=7 the exceptional set is bounded by 883 and for d=11 it is bounded by 8053.