Sum of squares length of real forms

2016/03/17 by Scheiderer, Claus
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1603.05430

Abstract

For n, d≥1 let p(n,2d) denote the smallest number p such that every sum of squares of forms of degree d in ℝ[x₁,…,x_n] is a sum of p squares. We establish lower bounds for these numbers that are considerably stronger than the bounds known so far. Combined with known upper bounds they give p(3,2d)∈\d+1, d+2\ in the ternary case. Assuming a conjecture of Iarrobino-Kanev on dimensions of tangent spaces to catalecticant varieties, we show that p(n,2d)∼ const⋅ d^(n-1)/2 for d→∞ and all n≥3. For ternary sextics and quaternary quartics we determine the exact value of the invariant, showing p(3,6)=4 and p(4,4)=5.

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