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First-order definability sums of squares

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6

$begingroup$

Let $K$ be a field. I am interested in when there can exist a first-order definition of the set
$$
Sigma K^2 := lbrace sum_{i=1}^n x_i^2 mid n in mathbb{N}, x_1, ldots, x_n in K rbrace
$$
in $K$ in the language of rings.

Clearly, if $K$ has finite Pythagoras number, then $Sigma K^2$ has an (existential) first-order definition in $K$. In fact, for every $n in mathbb{N}$ there is an existential first-order formula
$$
varphi_n(x) := exists x_1, ldots, x_n : x = sum_{i=1}^n x_i^2
$$
uniformly defining $Sigma K^2$ in all fields of Pythagoras number at most $n$.

Conversely, if $K$ has infinite Pythagoras number, then there can be no first-order formula uniformly defining $Sigma K'^2$ in all fields $K'$ elementarily equivalent to $K$. This follows from the compactness theorem.. However, this does not exclude the possibility of a field $K$ with infinite Pythagoras number and such that $Sigma K^2$ has a first-order definition in $K$ which does not carry over to fields elementarily equivalent to $K$.

So my broad question is to understand this problem better. More specifically, I would like to understand when $Sigma K^2$ is definable in a field $K$ of infinite Pythagoras number. Any example of a field with infinite Pythagoras number where you can prove or disprove that $Sigma K^2$ is (existentially) definable would already be very helpful.

field-theory first-order-logic model-theory

share|cite|improve this question

asked Mar 18 at 13:12

Bib-lostBib-lost

2,075629

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What are the standard examples of fields of infinite Pythagoras number?
  $endgroup$
  – Alessandro Codenotti
  Mar 19 at 19:45
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @AlessandroCodenotti It follows from a result by Cassels (although I cannot find a publicly accesible reference) that for any formally real field $K$ (i.e. -1 is not a sum of squares in $K$) one has that the Pythagoras number of $K(X)$ is at least one more than that of $K$. Hence the rational function field in infinitely many variables $K(X_n mid n in mathbb{N})$ has infinite Pythagoras number. If $K = mathbb{R}$, it is known that any finitely generated subfield of $mathbb{R}(X_n mid n in mathbb{N})$ has finite Pythagoras number.
  $endgroup$
  – Bib-lost
  Mar 19 at 21:29
  

  
  
  
  

add a comment | 

6

$begingroup$

Let $K$ be a field. I am interested in when there can exist a first-order definition of the set
$$
Sigma K^2 := lbrace sum_{i=1}^n x_i^2 mid n in mathbb{N}, x_1, ldots, x_n in K rbrace
$$
in $K$ in the language of rings.

Clearly, if $K$ has finite Pythagoras number, then $Sigma K^2$ has an (existential) first-order definition in $K$. In fact, for every $n in mathbb{N}$ there is an existential first-order formula
$$
varphi_n(x) := exists x_1, ldots, x_n : x = sum_{i=1}^n x_i^2
$$
uniformly defining $Sigma K^2$ in all fields of Pythagoras number at most $n$.

Conversely, if $K$ has infinite Pythagoras number, then there can be no first-order formula uniformly defining $Sigma K'^2$ in all fields $K'$ elementarily equivalent to $K$. This follows from the compactness theorem.. However, this does not exclude the possibility of a field $K$ with infinite Pythagoras number and such that $Sigma K^2$ has a first-order definition in $K$ which does not carry over to fields elementarily equivalent to $K$.

So my broad question is to understand this problem better. More specifically, I would like to understand when $Sigma K^2$ is definable in a field $K$ of infinite Pythagoras number. Any example of a field with infinite Pythagoras number where you can prove or disprove that $Sigma K^2$ is (existentially) definable would already be very helpful.

field-theory first-order-logic model-theory

share|cite|improve this question

asked Mar 18 at 13:12

Bib-lostBib-lost

2,075629

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What are the standard examples of fields of infinite Pythagoras number?
  $endgroup$
  – Alessandro Codenotti
  Mar 19 at 19:45
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @AlessandroCodenotti It follows from a result by Cassels (although I cannot find a publicly accesible reference) that for any formally real field $K$ (i.e. -1 is not a sum of squares in $K$) one has that the Pythagoras number of $K(X)$ is at least one more than that of $K$. Hence the rational function field in infinitely many variables $K(X_n mid n in mathbb{N})$ has infinite Pythagoras number. If $K = mathbb{R}$, it is known that any finitely generated subfield of $mathbb{R}(X_n mid n in mathbb{N})$ has finite Pythagoras number.
  $endgroup$
  – Bib-lost
  Mar 19 at 21:29
  

  
  
  
  

add a comment | 

$begingroup$

Let $K$ be a field. I am interested in when there can exist a first-order definition of the set
$$
Sigma K^2 := lbrace sum_{i=1}^n x_i^2 mid n in mathbb{N}, x_1, ldots, x_n in K rbrace
$$
in $K$ in the language of rings.

Clearly, if $K$ has finite Pythagoras number, then $Sigma K^2$ has an (existential) first-order definition in $K$. In fact, for every $n in mathbb{N}$ there is an existential first-order formula
$$
varphi_n(x) := exists x_1, ldots, x_n : x = sum_{i=1}^n x_i^2
$$
uniformly defining $Sigma K^2$ in all fields of Pythagoras number at most $n$.

Conversely, if $K$ has infinite Pythagoras number, then there can be no first-order formula uniformly defining $Sigma K'^2$ in all fields $K'$ elementarily equivalent to $K$. This follows from the compactness theorem.. However, this does not exclude the possibility of a field $K$ with infinite Pythagoras number and such that $Sigma K^2$ has a first-order definition in $K$ which does not carry over to fields elementarily equivalent to $K$.

So my broad question is to understand this problem better. More specifically, I would like to understand when $Sigma K^2$ is definable in a field $K$ of infinite Pythagoras number. Any example of a field with infinite Pythagoras number where you can prove or disprove that $Sigma K^2$ is (existentially) definable would already be very helpful.

field-theory first-order-logic model-theory

share|cite|improve this question

asked Mar 18 at 13:12

Bib-lostBib-lost

2,075629

$endgroup$

share|cite|improve this question

asked Mar 18 at 13:12

Bib-lostBib-lost

2,075629

share|cite|improve this question

asked Mar 18 at 13:12

Bib-lostBib-lost

2,075629

asked Mar 18 at 13:12

Bib-lostBib-lost

2,075629

asked Mar 18 at 13:12

Bib-lostBib-lost

2,075629