Minimization of hypersurfaces

TL;DR

This paper establishes bounds on weight vectors for minimizing hypersurfaces over integers and primes, and introduces efficient algorithms for minimizing plane curves and cubic surfaces, implemented in Magma.

Contribution

It provides explicit bounds on weight vectors for hypersurface minimization and develops practical algorithms for plane curves and cubic surfaces.

Findings

Bound of $[0,w_1,w_2,...,w_n]$ with $w_n \\le 2 n d^{n-1}$ for minimization

Improved bound to $d$ when $n=2$ for plane curves

Algorithms for minimization of ternary forms and cubic surfaces in Magma

Abstract

Let $F \in \mathbb{Z}[x_0, \ldots, x_n]$ be homogeneous of degree $d$ and assume that $F$ is not a `nullform', i.e., there is an invariant $I$ of forms of degree $d$ in $n+1$ variables such that $I(F) \neq 0$. Equivalently, $F$ is semistable in the sense of Geometric Invariant Theory. Minimizing $F$ at a prime $p$ means to produce $T \in \operatorname{Mat}(n+1, \mathbb{Z}) \cap \operatorname{GL}(n+1, \mathbb{Q})$ and $e \in \mathbb{Z}_{\ge 0}$ such that $F_1 = p^{-e} F([x_0, \ldots, x_n] \cdot T)$ has integral coefficients and $v_p(I(F_1))$ is minimal among all such $F_1$. Following Koll\'ar, the minimization process can be described in terms of applying weight vectors $w \in \mathbb{Z}_{\ge 0}^{n+1}$ to $F$. We show that for any dimension $n$ and degree $d$, there is a complete set of weight vectors consisting of $[0,w_1,w_2,\dots,w_n]$ with $0 \le w_1 \le w_2 \le \dots \le w_n \le 2 n d^{n-1}$. When $n = 2$, we improve the bound to $d$. This answers a question raised by Koll\'ar. These results are valid in a more general context, replacing $\mathbb{Z}$ and $p$ by a PID $R$ and a prime element of $R$. Based on this result and a further study of the minimization process in the planar case $n = 2$, we devise an efficient minimization algorithm for ternary forms (equivalently, plane curves) of arbitrary degree $d$. We also describe a similar algorithm that allows to minimize (and reduce) cubic surfaces. The algorithms are available in the computer algebra system Magma.