Sums of squares in fields - Karim Becher (Università di Antwerp, Belgio)
Minicorso in ambito Programma Erasmus
Dettagli dell'evento
Quando
04/04/2017
dalle 15:30 alle 17:30
dalle 15:30 alle 17:30
Dove
Dipartimento di Matematica e Informatica
Persona di riferimento
Claudia Menini
Calendario:
- martedì 4 aprile ore 15.30 aula 4
- mercoledì 5 aprile ore 15.30 aula 4
- giovedì 6 aprile ore 15.30 aula 5
- venerdì 7 aprile ore 10.30 aula 1
Programma:
- I Sums of squares in number theory
- II Sums of squares in real algebra
- III Basic quadratic form theory over fields
- IV Local-global principles for quadratic forms
Abstract:
In this course I want to touch on several aspects of sums of squares and their role in the development of modern algebra. The focus will be on sums of squares in commutative rings and in particular in fields.
Some basics of the theory of quadratic forms over fields will be introduced. The course will only assume some general algebraic knowledge on bachelor level.
The study of sums of squares is an old topic in number theory and algebra.
In Brahmagupta’s book an identity is given for writing a product of two sums of two squares again as a sum of two squares.
In modern terms it can be obtained by using the norm form for the ring of Gaussian integers, or more generally for the quadratic extension of a given commutative ring by adjoining a square root of -1.
Euler gave a similar identity for products of sums of four squares, which can likewise be viewed as the norm form of an algebraic structure, the quaternions.
He probably did this with the goal to prove that every positive integer is a sum of four squares, which was one of Fermat’s (correct) statements.
This way the problem had been reduced to showing the claim for prime numbers, and on this basis the proof was later completed by Lagrange.
Later Degen and Cayley found independently a similar identity for sums of eight squares and a related algebraic structure, the octonions.
On the other hand, attempts to find such an identity for sums of 16 squares failed and in the end Hurwitz showed that such an identity cannot exist.
More generally, the problem is to express a product of two quadratic forms (homogeneous polynomials of degree two) as a quadratic form applied to bilinear expressions in the variables.
If this is possible then a solution to this problem is called a composition formula. Using Clifford algebras one can explain why composition formulae do not exist for certain triples of forms, just by their dimensions.
Many problems of an analytic flavour can be described by asking whether a certain real valued function takes only non-negative values.
An obviously sufficient condition is that the function can be written as a sum of squares of other functions.
For rational functions (fractions of polynomials), Hilbert asked in his 17th problem whether the converse is true, namely whether any rational function taking only non-negative values can be written as a sum of squares of rational functions.
The positive answer was found by Artin in 1927 and it opened the way to a new research area, called real algebra.
Here different versions of the problem can be considered, for example, for a polynomial taking only non-negative values, one can ask whether it can be written as a sum of squares of polynomials.
This is generally not true. But also the number of squares in such an expression can be studied. This gives rise to the definition of the Pythagoras number of a commutative ring, the smallest number p such that every sum of squares is equal to a sum of p squares.
For a field we may ask whether -1 can be written as a sum of squares in it and if so, how many squares are needed. The smallest such number s, if it exists, is called the level of the field.
Pfister’s showed that the level of a field where -1 is a sum of squares is always a 2-power and that on the other hand every 2-power is the level of some field.
The proofs of these results give a nice insight into the topics of quadratic form theory over fields.
A main feature of 20th century algebra, number theory and geometry is the local-global analysis of problems.
It started with Hensel’s construction of the p-adic integers. It is not too difficult to see that polynomials with integer coefficients can have local solutions over every p-adic completion but no solution over the integers, but specifically for quadratic forms, by the Hasse-Minkowski Theorem, the local-global way of reasoning for finding (nontrivial) solutions works. Unfortunately, the same fails for quadratic forms over most other types of fields (when taking completions with respect to discrete valuations).
Recently, however, it was proven for certain types of function fields of arithmetic curves that such a local-global principle holds. In my own research I am studying some of these situations with the particular interest for the study of sums of squares, say, over the function field of a curve over R((t)), the field of formal Laurent series with real coefficients. I will discuss some of these results in relation to the study of valuations on these fields and give some examples.
Some basics of the theory of quadratic forms over fields will be introduced. The course will only assume some general algebraic knowledge on bachelor level.
The study of sums of squares is an old topic in number theory and algebra.
In Brahmagupta’s book an identity is given for writing a product of two sums of two squares again as a sum of two squares.
In modern terms it can be obtained by using the norm form for the ring of Gaussian integers, or more generally for the quadratic extension of a given commutative ring by adjoining a square root of -1.
Euler gave a similar identity for products of sums of four squares, which can likewise be viewed as the norm form of an algebraic structure, the quaternions.
He probably did this with the goal to prove that every positive integer is a sum of four squares, which was one of Fermat’s (correct) statements.
This way the problem had been reduced to showing the claim for prime numbers, and on this basis the proof was later completed by Lagrange.
Later Degen and Cayley found independently a similar identity for sums of eight squares and a related algebraic structure, the octonions.
On the other hand, attempts to find such an identity for sums of 16 squares failed and in the end Hurwitz showed that such an identity cannot exist.
More generally, the problem is to express a product of two quadratic forms (homogeneous polynomials of degree two) as a quadratic form applied to bilinear expressions in the variables.
If this is possible then a solution to this problem is called a composition formula. Using Clifford algebras one can explain why composition formulae do not exist for certain triples of forms, just by their dimensions.
Many problems of an analytic flavour can be described by asking whether a certain real valued function takes only non-negative values.
An obviously sufficient condition is that the function can be written as a sum of squares of other functions.
For rational functions (fractions of polynomials), Hilbert asked in his 17th problem whether the converse is true, namely whether any rational function taking only non-negative values can be written as a sum of squares of rational functions.
The positive answer was found by Artin in 1927 and it opened the way to a new research area, called real algebra.
Here different versions of the problem can be considered, for example, for a polynomial taking only non-negative values, one can ask whether it can be written as a sum of squares of polynomials.
This is generally not true. But also the number of squares in such an expression can be studied. This gives rise to the definition of the Pythagoras number of a commutative ring, the smallest number p such that every sum of squares is equal to a sum of p squares.
For a field we may ask whether -1 can be written as a sum of squares in it and if so, how many squares are needed. The smallest such number s, if it exists, is called the level of the field.
Pfister’s showed that the level of a field where -1 is a sum of squares is always a 2-power and that on the other hand every 2-power is the level of some field.
The proofs of these results give a nice insight into the topics of quadratic form theory over fields.
A main feature of 20th century algebra, number theory and geometry is the local-global analysis of problems.
It started with Hensel’s construction of the p-adic integers. It is not too difficult to see that polynomials with integer coefficients can have local solutions over every p-adic completion but no solution over the integers, but specifically for quadratic forms, by the Hasse-Minkowski Theorem, the local-global way of reasoning for finding (nontrivial) solutions works. Unfortunately, the same fails for quadratic forms over most other types of fields (when taking completions with respect to discrete valuations).
Recently, however, it was proven for certain types of function fields of arithmetic curves that such a local-global principle holds. In my own research I am studying some of these situations with the particular interest for the study of sums of squares, say, over the function field of a curve over R((t)), the field of formal Laurent series with real coefficients. I will discuss some of these results in relation to the study of valuations on these fields and give some examples.
