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Sidney Raffer
Sidney Raffer
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Let's say that a discretely ordered ring has rank 1 if it has elements greater than any integer, and for any two such elements $x<y$ there is an integer $n$ such that $x^n>y$.

Question: Let $f(\bar{x})$ be a polynomial in any number of variables with integer coefficients that has a zero in at least one discretely ordered ring. Must $f$ have a zero in a discretely ordered ring of rank 1?

Let's say that a discretely ordered ring has rank 1 if it has elements greater than any integer, and for any two such elements $x<y$ there is an integer $n$ such that $x^n>y$.

Question: Let $f(\bar{x})$ be a polynomial with integer coefficients that has a zero in at least one discretely ordered ring. Must $f$ have a zero in a discretely ordered ring of rank 1?

Let's say that a discretely ordered ring has rank 1 if it has elements greater than any integer, and for any two such elements $x<y$ there is an integer $n$ such that $x^n>y$.

Question: Let $f(\bar{x})$ be a polynomial in any number of variables with integer coefficients that has a zero in at least one discretely ordered ring. Must $f$ have a zero in a discretely ordered ring of rank 1?

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Sidney Raffer
Sidney Raffer
  • 6.2k
  • 1
  • 27
  • 42

Let's say that a discretely ordered ring has rank 1 if it has elements greater than any integer, and for any two such elements $x$ and $y$$x<y$ there are integersis an integer $n$ and $m$ such that $x^n>y$ and $y^m>x$.

Question: Let $f(\bar{x})$ be a polynomial with integer coefficients that has a zero in at least one discretely ordered ring. Must $f$ have a zero in a discretely ordered ring of rank 1?

Let's say that a discretely ordered ring has rank 1 if it has elements greater than any integer, and for any two such elements $x$ and $y$ there are integers $n$ and $m$ such that $x^n>y$ and $y^m>x$.

Question: Let $f(\bar{x})$ be a polynomial with integer coefficients that has a zero in at least one discretely ordered ring. Must $f$ have a zero in a discretely ordered ring of rank 1?

Let's say that a discretely ordered ring has rank 1 if it has elements greater than any integer, and for any two such elements $x<y$ there is an integer $n$ such that $x^n>y$.

Question: Let $f(\bar{x})$ be a polynomial with integer coefficients that has a zero in at least one discretely ordered ring. Must $f$ have a zero in a discretely ordered ring of rank 1?

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Sidney Raffer
Sidney Raffer
  • 6.2k
  • 1
  • 27
  • 42

Zeros of polynomials in discretely ordered rings

Let's say that a discretely ordered ring has rank 1 if it has elements greater than any integer, and for any two such elements $x$ and $y$ there are integers $n$ and $m$ such that $x^n>y$ and $y^m>x$.

Question: Let $f(\bar{x})$ be a polynomial with integer coefficients that has a zero in at least one discretely ordered ring. Must $f$ have a zero in a discretely ordered ring of rank 1?