First-order definability sums of squares The Next CEO of Stack OverflowDefinability in exponential fields assuming quasiminimalityWhy isn't there a first-order theory of well order?Extending the language in Henkin style completeness proof for first-order logicDefinability of a setIs metric (Cauchy) completeness “outside the realm” of first order logic?Do canonical Skolem hulls witness first order definable well-orders?First-order properties and models of $mathbbQ$Is $mathbb Q$ definable in $bar mathbb Q$ ?First-order definability of structures of at least $n$ elementsFirst-Order Definability of finite structures (negative result)
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First-order definability sums of squares
The Next CEO of Stack OverflowDefinability in exponential fields assuming quasiminimalityWhy isn't there a first-order theory of well order?Extending the language in Henkin style completeness proof for first-order logicDefinability of a setIs metric (Cauchy) completeness “outside the realm” of first order logic?Do canonical Skolem hulls witness first order definable well-orders?First-order properties and models of $mathbbQ$Is $mathbb Q$ definable in $bar mathbb Q$ ?First-order definability of structures of at least $n$ elementsFirst-Order Definability of finite structures (negative result)
$begingroup$
Let $K$ be a field. I am interested in when there can exist a first-order definition of the set
$$
Sigma K^2 := lbrace sum_i=1^n x_i^2 mid n in mathbbN, x_1, ldots, x_n in K rbrace
$$
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 $n in mathbbN$ 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
asked Mar 18 at 13:12
Bib-lostBib-lost
2,075629
$endgroup$
add a comment |
$begingroup$
Let $K$ be a field. I am interested in when there can exist a first-order definition of the set
$$
Sigma K^2 := lbrace sum_i=1^n x_i^2 mid n in mathbbN, x_1, ldots, x_n in K rbrace
$$
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 $n in mathbbN$ 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
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 $K(X_n mid n in mathbbN)$ has infinite Pythagoras number. If $K = mathbbR$, it is known that any finitely generated subfield of $mathbbR(X_n mid n in mathbbN)$ 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
$$
Sigma K^2 := lbrace sum_i=1^n x_i^2 mid n in mathbbN, x_1, ldots, x_n in K rbrace
$$
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 $n in mathbbN$ 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
asked Mar 18 at 13:12
Bib-lostBib-lost
2,075629
$endgroup$
Let $K$ be a field. I am interested in when there can exist a first-order definition of the set
$$
Sigma K^2 := lbrace sum_i=1^n x_i^2 mid n in mathbbN, x_1, ldots, x_n in K rbrace
$$
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 $n in mathbbN$ 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
field-theory first-order-logic model-theory
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
asked Mar 18 at 13:12
Bib-lostBib-lost
2,075629
asked Mar 18 at 13:12
Bib-lostBib-lost
2,075629
2,075629
$begingroup$
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 $K(X_n mid n in mathbbN)$ has infinite Pythagoras number. If $K = mathbbR$, it is known that any finitely generated subfield of $mathbbR(X_n mid n in mathbbN)$ has finite Pythagoras number.
$endgroup$
– Bib-lost
Mar 19 at 21:29
add a comment |
$begingroup$
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 $K(X_n mid n in mathbbN)$ has infinite Pythagoras number. If $K = mathbbR$, it is known that any finitely generated subfield of $mathbbR(X_n mid n in mathbbN)$ has finite Pythagoras number.
$endgroup$
– Bib-lost
Mar 19 at 21:29
$begingroup$
What are the standard examples of fields of infinite Pythagoras number?
$endgroup$
– Alessandro Codenotti
Mar 19 at 19:45
$begingroup$
What are the standard examples of fields of infinite Pythagoras number?
$endgroup$
– Alessandro Codenotti
Mar 19 at 19:45
1
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 $K(X_n mid n in mathbbN)$ has infinite Pythagoras number. If $K = mathbbR$, it is known that any finitely generated subfield of $mathbbR(X_n mid n in mathbbN)$ has finite Pythagoras number.
$endgroup$
– Bib-lost
Mar 19 at 21:29
$begingroup$
@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 $K(X_n mid n in mathbbN)$ has infinite Pythagoras number. If $K = mathbbR$, it is known that any finitely generated subfield of $mathbbR(X_n mid n in mathbbN)$ has finite Pythagoras number.
$endgroup$
– Bib-lost
Mar 19 at 21:29
add a comment |
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$begingroup$
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 $K(X_n mid n in mathbbN)$ has infinite Pythagoras number. If $K = mathbbR$, it is known that any finitely generated subfield of $mathbbR(X_n mid n in mathbbN)$ has finite Pythagoras number.
$endgroup$
– Bib-lost
Mar 19 at 21:29