Papers › Minimization of hypersurfaces
Minimization of hypersurfaces
Andreas-Stephan Elsenhans, Michael Stoll
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Let F ∈ℤ[x₀, …, xₙ] 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) ≠0. Equivalently, F is semistable in the sense of Geometric Invariant Theory. Minimizing F at a prime p means to produce T ∈Mat(n+1, ℤ) ∩GL(n+1, ℚ) and e ∈ℤ_(≥0) such that F₁ = p⁻ᵉ F([x₀, …, xₙ] ·T) has integral coefficients and vₚ(I(F₁)) is minimal among all such F₁. Following Koll\'ar, the minimization process can be described in terms of applying weight vectors w ∈ℤ_(≥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,w₁,w₂,…,wₙ] with 0 ≤w₁ ≤w₂ ≤…≤wₙ ≤2 n dⁿ⁻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 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.
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