Publikation: Sum of squares length of real forms
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2017
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Mathematische Zeitschrift. 2017, 286(1-2), pp. 559-570. ISSN 0025-5874. eISSN 1432-1823. Available under: doi: 10.1007/s00209-016-1773-z
Zusammenfassung
For n,d≥1 let p(n, 2d) denote the smallest number p such that every sum of squares of degree d forms in R[x1,…,xn] is a sum of p squares. We establish lower bounds for p(n, 2d) 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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510 Mathematik
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SCHEIDERER, Claus, 2017. Sum of squares length of real forms. In: Mathematische Zeitschrift. 2017, 286(1-2), pp. 559-570. ISSN 0025-5874. eISSN 1432-1823. Available under: doi: 10.1007/s00209-016-1773-zBibTex
@article{Scheiderer2017squar-39241,
year={2017},
doi={10.1007/s00209-016-1773-z},
title={Sum of squares length of real forms},
number={1-2},
volume={286},
issn={0025-5874},
journal={Mathematische Zeitschrift},
pages={559--570},
author={Scheiderer, Claus}
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<dcterms:abstract xml:lang="eng">For n,d≥1 let p(n, 2d) denote the smallest number p such that every sum of squares of degree d forms in R[x1,…,xn] is a sum of p squares. We establish lower bounds for p(n, 2d) 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.</dcterms:abstract>
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