Let $F(x,y) = a_4 x^4 + a_3 x^3 y + a_2 x^2 y^2 + a_1 xy^3 + a_0 y^4$ be a binary quartic form with real coefficients. It is well-known the action of $\operatorname{GL}_2^{(\pm)} (\mathbb{R})$ consisting of real invertible matrices with determininant $\pm 1$ on the real vector space $V_\mathbb{R}$ of binary quartic forms under substitution has two invariants, usually denoted by $I$ and $J$, which are given by

$$\displaystyle I(F) = 12a_4 a_0 - 3a_3 a_1 + a_2^2$$ and $$\displaystyle J(F) = 72 a_4 a_2 a_0 + 9 a_3 a_2 a_1 - 27 a_4 a_1^2 - 27 a_0 a_3^2 - 2a_2^3.$$

In their paper Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves, Bhargava and Shankar introduced the following $\operatorname{GL}_2^{(\pm)}(\mathbb{R})$-invariant height on $V_\mathbb{R}$: $H(F) = \max\{|I(F)|^3, J(F)^2/4\}$. They called this the naive height, but we shall call it the `Bhargava-Shankar height' for the sake of this post in case one may use the phrase 'naive height' to refer the more simple box height given by the maximum of the coefficients of $F$ (this is mostly for the case when $F$ has integer coefficients).

One might ask why is it natural to consider such a height. This is likely motivated by their goal of estimating the $2$-Selmer rank of elliptic curves $E/\mathbb{Q}$. To do so, one essentially has to count everywhere locally soluble $\operatorname{PGL}_2(\mathbb{Q})$ orbits of integral binary quartic forms with respect to some height, and then count elliptic curves over $\mathbb{Q}$ with respect to a compatible height. Thus, the way to count elliptic curves is a strong motivator for the definition of the Bhargava-Shankar height. An elliptic curve $E/\mathbb{Q}$ can be uniquely given in terms of the Weierstrass model, namely there exist integers $A,B$ such that $$\displaystyle E: y^2 = x^3 + Ax + B,$$

and this model is unique if whenever $p^4 | A$ for some prime $p$, we have $p^6 \nmid B$. Moreover, $A,B$ are the values of natural invariants of the cubic polynomial $x^3 + Ax + B$ under the substitution action by the group of lower triangular matrices with 1's on the diagonal. It is therefore natural to consider the height on elliptic curves given by

$$\displaystyle H(E) = H(E(A,B)) = \max\{|A|^3, B^2\}.$$

One of the major innovations of Bhargava and Shankar's paper is their construction of an invariant preserving map between a binary quartic form and the corresponding elliptic curve. Thus, the two heights are compatible and one can directly compare the count of ELS binary quartic forms and elliptic curves over the rationals.

However, the picture is not so clear for higher degrees. Polynomials of every degree have invariants under substitution by $\operatorname{GL}_2^{(\pm)} (\mathbb{R})$, but it is not clear how to arrange them to obtain a reasonable height. Of course, the most canonical way to count polynomials is by their discriminant, which is always an invariant itself. However, as far as I know, this has not been done for degree $d > 3$.

So my question is, are there other natural objects associated with the set of degree $d$ polynomials for some $d > 4$ which carries with it a natural height which can then be interpreted for degree $d$ polynomials in terms of its invariants, much like the Bhargava-Shankar height for the case of binary quartic forms?


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