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fixed my mistake, noticed by J.S.
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Bobby Grizzard
Bobby Grizzard
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Let $f(X,t_1,\dots,t_s)$ be an irreducible polynomial with coefficients in $\mathcal{O}_K$, the ring of integers of a number field $K$. By work of S. D. Cohen (http://plms.oxfordjournals.org/content/s3-43/2/227.full.pdf) I know that the number of specializations $\vec t = (t_1,\dots, t_s)$ in $(\mathcal{O}_K)^s$ of height at most $N$, such that the specialization remainsdoes not remain irreducible (indeed retainsor even does not retain the original Galois group of $f$) is $O(N^{s-1/2}\log(N))$, with the implied constant depending only on $f, s,$ and $K$.

  1. Is this the best result known, in terms of the order of growth in $N$?
  2. Is it conjectured that the order can be improved, say to $O(N^{s-1})$? (Maybe not for $s=1?$)
  3. Regarding the first two questions, is more known/conjectured in the special case that $f$ is a polynomial in $X$ where precisely $s$ of the coefficients are allowed to vary in $\mathcal{O}_K$? (i.e. $f \in \mathcal{O}_K[t_1,\dots,t_s][X]$, with each coefficient having degree 0 or 1 in $t_1,\dots,t_s$? In this special case, the best results I know of, also due to Cohen (http://www.projecteuclid.org/euclid.ijm/1256048323) are of the same order as the more general results, except for $s=1$, where the log factor can be omitted.)
  4. As above, but in the special case $K=\mathbb{Q}$?

Let $f(X,t_1,\dots,t_s)$ be an irreducible polynomial with coefficients in $\mathcal{O}_K$, the ring of integers of a number field $K$. By work of S. D. Cohen (http://plms.oxfordjournals.org/content/s3-43/2/227.full.pdf) I know that the number of specializations $\vec t = (t_1,\dots, t_s)$ in $(\mathcal{O}_K)^s$ of height at most $N$, such that the specialization remains irreducible (indeed retains the original Galois group of $f$) is $O(N^{s-1/2}\log(N))$, with the implied constant depending only on $f, s,$ and $K$.

  1. Is this the best result known, in terms of the order of growth in $N$?
  2. Is it conjectured that the order can be improved, say to $O(N^{s-1})$? (Maybe not for $s=1?$)
  3. Regarding the first two questions, is more known/conjectured in the special case that $f$ is a polynomial in $X$ where precisely $s$ of the coefficients are allowed to vary in $\mathcal{O}_K$? (i.e. $f \in \mathcal{O}_K[t_1,\dots,t_s][X]$, with each coefficient having degree 0 or 1 in $t_1,\dots,t_s$? In this special case, the best results I know of, also due to Cohen (http://www.projecteuclid.org/euclid.ijm/1256048323) are of the same order as the more general results, except for $s=1$, where the log factor can be omitted.)
  4. As above, but in the special case $K=\mathbb{Q}$?

Let $f(X,t_1,\dots,t_s)$ be an irreducible polynomial with coefficients in $\mathcal{O}_K$, the ring of integers of a number field $K$. By work of S. D. Cohen (http://plms.oxfordjournals.org/content/s3-43/2/227.full.pdf) I know that the number of specializations $\vec t = (t_1,\dots, t_s)$ in $(\mathcal{O}_K)^s$ of height at most $N$, such that the specialization does not remain irreducible (or even does not retain the original Galois group of $f$) is $O(N^{s-1/2}\log(N))$, with the implied constant depending only on $f, s,$ and $K$.

  1. Is this the best result known, in terms of the order of growth in $N$?
  2. Is it conjectured that the order can be improved, say to $O(N^{s-1})$? (Maybe not for $s=1?$)
  3. Regarding the first two questions, is more known/conjectured in the special case that $f$ is a polynomial in $X$ where precisely $s$ of the coefficients are allowed to vary in $\mathcal{O}_K$? (i.e. $f \in \mathcal{O}_K[t_1,\dots,t_s][X]$, with each coefficient having degree 0 or 1 in $t_1,\dots,t_s$? In this special case, the best results I know of, also due to Cohen (http://www.projecteuclid.org/euclid.ijm/1256048323) are of the same order as the more general results, except for $s=1$, where the log factor can be omitted.)
  4. As above, but in the special case $K=\mathbb{Q}$?
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Bobby Grizzard
Bobby Grizzard
  • 1.5k
  • 1
  • 10
  • 21

The best possible density in Hilbert's Irreducibility Theorem

Let $f(X,t_1,\dots,t_s)$ be an irreducible polynomial with coefficients in $\mathcal{O}_K$, the ring of integers of a number field $K$. By work of S. D. Cohen (http://plms.oxfordjournals.org/content/s3-43/2/227.full.pdf) I know that the number of specializations $\vec t = (t_1,\dots, t_s)$ in $(\mathcal{O}_K)^s$ of height at most $N$, such that the specialization remains irreducible (indeed retains the original Galois group of $f$) is $O(N^{s-1/2}\log(N))$, with the implied constant depending only on $f, s,$ and $K$.

  1. Is this the best result known, in terms of the order of growth in $N$?
  2. Is it conjectured that the order can be improved, say to $O(N^{s-1})$? (Maybe not for $s=1?$)
  3. Regarding the first two questions, is more known/conjectured in the special case that $f$ is a polynomial in $X$ where precisely $s$ of the coefficients are allowed to vary in $\mathcal{O}_K$? (i.e. $f \in \mathcal{O}_K[t_1,\dots,t_s][X]$, with each coefficient having degree 0 or 1 in $t_1,\dots,t_s$? In this special case, the best results I know of, also due to Cohen (http://www.projecteuclid.org/euclid.ijm/1256048323) are of the same order as the more general results, except for $s=1$, where the log factor can be omitted.)
  4. As above, but in the special case $K=\mathbb{Q}$?