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Let $D_4$ denote the (isomorphism class of) dihedral group of 8 elements. For a given binary quartic form $f$ with integer coefficients, define $\text{Gal}(f)$ to be the Galois group of the Galois closure of $f(x,1)$ and $H(f) = \max\{|I(f)|^3, J(f)^2/4\}$ to be the naive height of $f$, defined by Bhargava and Shankar (see references below). Put $\nu(f)$ for the $\text{GL}_2(\mathbb{Z})$-equivalence class of $f$, under the action of $\text{GL}_2(\mathbb{Z})$ on the space of binary quartic forms. Consider the counting function

$$\displaystyle N(f; D_4, Z) = \#\{\nu(f): f \in \mathbb{Z}[x], \deg f = 4, H(f) < Z, \text{Gal}(f) \subset D_4 \}.$$

Is there a known asymptotic formula for $N(f; D_4, Z)$, or at least an asymptotic lower bound expected to be of the correct order of magnitude?

There are several related results. For example, if one is to count the number of quartic fields with Galois group $D_4$ instead, then it was shown by Cohen, Diaz y Diaz, and Olivier that there are roughly $cZ$ many such fields with absolute discriminant up to $Z$, where $c \approx 0.0523\cdots$. One should note that the density of quartic fields with Galois group $D_4$ among all quartic fields is positive, as was shown by Bhargava. This is in contrast to the case with polynomials, where $100\%$ of all quartic polynomials have Galois group $S_4$. It was further shown by Bhargava and Shankar that there are at most $O_{\varepsilon}\left(Z^{\frac{3}{4} + \epsilon}\right)$ number of quartic polynomials of "naive height" up to $Z$, where as the number of quartic polynomials of naive height up to $Z$ is $Z^{\frac{5}{6}}$. The naive height is defined using the $I,J$ invariants of quartic polynomials and is designed to mimic the discriminant but is easier to deal with. The order of magnitude should be the same for both counting methods.

References:

H. Cohen, F. Diaz y Diaz, and M. Olivier, Enumerating quartic dihedral extensions of $\mathbb{Q}$, Compositio Math. 133 (2002), 65–93.

M. Bhargava, The density of discriminants of quartic rings and fields, Annals of Mathematics 162 (2005), 1031-1063.

M. Bhargava, A. Shankar, Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves, Annals of Mathematics 181 (2015), 191-242.

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    $\begingroup$ This may be a dumb question, but why is the quantity you want to estimate finite? $\endgroup$ – Igor Rivin Feb 16 '16 at 20:20
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    $\begingroup$ @IgorRivin: I meant to count only up to $\text{GL}_2(\mathbb{Z})$ equivalence, since there are infinitely many polynomials with a given discriminant (by applying $\text{GL}_2(\mathbb{Z})$-action). Of course, they will all have the same Galois group. Thanks for pointing this out! $\endgroup$ – Stanley Yao Xiao Feb 16 '16 at 20:25

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