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Fixes the definition of $S_{d, H}$: $f$ is required to be non-zero
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Luc Guyot
Luc Guyot
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Let $f(x)=a_nx^n+\cdots+a_0 \in \mathbb{Z}[x]$ be an integer polynomial in one variable. Recall that the height $H(f):=\textrm{max}\,|a_n|$ is the largest coefficient. Consider the set of polynomials of fixed degree which divide a non-zero polynomial of bounded height: $$S_{d,H}:=\{g\in\mathbb{Z}[x]\,\big{|}\,\deg{g}=d\textrm{ and }g\mid f\textrm{ for some }f\textrm{ with }H(f)\leq H\}.$$$$S_{d,H}:=\{g\in\mathbb{Z}[x]\,\big{|}\,\deg{g}=d\textrm{ and }g\mid f\textrm{ for some non-zero}f\textrm{ with }H(f)\leq H\}.$$ Crucially, there are no conditions on the degree of $f$. Is $S_{d,H}$ a finite set for all $d$ and $H$? If so, is there a known bound in the literature for $\textrm{max}_{g\in S_{d,H}} H(g)$ in terms of $H$ and $d$?

Let $f(x)=a_nx^n+\cdots+a_0 \in \mathbb{Z}[x]$ be an integer polynomial in one variable. Recall that the height $H(f):=\textrm{max}\,|a_n|$ is the largest coefficient. Consider the set of polynomials of fixed degree which divide a polynomial of bounded height: $$S_{d,H}:=\{g\in\mathbb{Z}[x]\,\big{|}\,\deg{g}=d\textrm{ and }g\mid f\textrm{ for some }f\textrm{ with }H(f)\leq H\}.$$ Crucially, there are no conditions on the degree of $f$. Is $S_{d,H}$ a finite set for all $d$ and $H$? If so, is there a known bound in the literature for $\textrm{max}_{g\in S_{d,H}} H(g)$ in terms of $H$ and $d$?

Let $f(x)=a_nx^n+\cdots+a_0 \in \mathbb{Z}[x]$ be an integer polynomial in one variable. Recall that the height $H(f):=\textrm{max}\,|a_n|$ is the largest coefficient. Consider the set of polynomials of fixed degree which divide a non-zero polynomial of bounded height: $$S_{d,H}:=\{g\in\mathbb{Z}[x]\,\big{|}\,\deg{g}=d\textrm{ and }g\mid f\textrm{ for some non-zero}f\textrm{ with }H(f)\leq H\}.$$ Crucially, there are no conditions on the degree of $f$. Is $S_{d,H}$ a finite set for all $d$ and $H$? If so, is there a known bound in the literature for $\textrm{max}_{g\in S_{d,H}} H(g)$ in terms of $H$ and $d$?

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Philip Engel
Philip Engel
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Let $f(x)=a_nx^n+\cdots+a_0 \in \mathbb{Z}[x]$ be an integer polynomial in one variable. Recall that the height $H(f):=\textrm{max}\,|a_n|$ is the largest coefficient. Consider the set of polynomials of fixed degree which divide a polynomial of bounded height: $$S_{d,H}:=\{g\in\mathbb{Z}[x]\,\big{|}\,\deg{g}=d\textrm{ and }g\mid f\textrm{ for some }f\textrm{ with }H(f)\leq H\}.$$ Crucially, there are no conditions on the degree of $f$. Is $S_{d,H}$ a finite set for all $d$ and $H$? If so, is there a known bound in the literature for $\textrm{max}_{g\in S_{d,h}} H(g)$$\textrm{max}_{g\in S_{d,H}} H(g)$ in terms of $H$ and $d$?

Let $f(x)=a_nx^n+\cdots+a_0 \in \mathbb{Z}[x]$ be an integer polynomial in one variable. Recall that the height $H(f):=\textrm{max}\,|a_n|$ is the largest coefficient. Consider the set of polynomials of fixed degree which divide a polynomial of bounded height: $$S_{d,H}:=\{g\in\mathbb{Z}[x]\,\big{|}\,\deg{g}=d\textrm{ and }g\mid f\textrm{ for some }f\textrm{ with }H(f)\leq H\}.$$ Crucially, there are no conditions on the degree of $f$. Is $S_{d,H}$ a finite set for all $d$ and $H$? If so, is there a known bound in the literature for $\textrm{max}_{g\in S_{d,h}} H(g)$ in terms of $H$ and $d$?

Let $f(x)=a_nx^n+\cdots+a_0 \in \mathbb{Z}[x]$ be an integer polynomial in one variable. Recall that the height $H(f):=\textrm{max}\,|a_n|$ is the largest coefficient. Consider the set of polynomials of fixed degree which divide a polynomial of bounded height: $$S_{d,H}:=\{g\in\mathbb{Z}[x]\,\big{|}\,\deg{g}=d\textrm{ and }g\mid f\textrm{ for some }f\textrm{ with }H(f)\leq H\}.$$ Crucially, there are no conditions on the degree of $f$. Is $S_{d,H}$ a finite set for all $d$ and $H$? If so, is there a known bound in the literature for $\textrm{max}_{g\in S_{d,H}} H(g)$ in terms of $H$ and $d$?

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Philip Engel
Philip Engel
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Factors of polynomials of bounded height

Let $f(x)=a_nx^n+\cdots+a_0 \in \mathbb{Z}[x]$ be an integer polynomial in one variable. Recall that the height $H(f):=\textrm{max}\,|a_n|$ is the largest coefficient. Consider the set of polynomials of fixed degree which divide a polynomial of bounded height: $$S_{d,H}:=\{g\in\mathbb{Z}[x]\,\big{|}\,\deg{g}=d\textrm{ and }g\mid f\textrm{ for some }f\textrm{ with }H(f)\leq H\}.$$ Crucially, there are no conditions on the degree of $f$. Is $S_{d,H}$ a finite set for all $d$ and $H$? If so, is there a known bound in the literature for $\textrm{max}_{g\in S_{d,h}} H(g)$ in terms of $H$ and $d$?