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Pythagoras Numbers for Ternary Forms

Grigoriy Blekherman, Alex Dunbar, Rainer Sinn

Abstract

We study the Pythagoras numbers $py(3,2d)$ of real ternary forms, defined for each degree $2d$ as the minimal number $r$ such that every degree $2d$ ternary form which is a sum of squares can be written as the sum of at most $r$ squares of degree $d$ forms. Scheiderer showed that $d+1\leq py(3,2d)\leq d+2$. We show that $py(3,2d) = d+1$ for $2d = 8,10,12$. The main technical tool is Diesel's characterization of height 3 Gorenstein algebras.

Pythagoras Numbers for Ternary Forms

Abstract

We study the Pythagoras numbers of real ternary forms, defined for each degree as the minimal number such that every degree ternary form which is a sum of squares can be written as the sum of at most squares of degree forms. Scheiderer showed that . We show that for . The main technical tool is Diesel's characterization of height 3 Gorenstein algebras.

Paper Structure

This paper contains 20 sections, 19 theorems, 46 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Outline of the paper
  3. Notation and Conventions
  4. Pythagoras Number to Gorenstein Ideals
  5. Minimizing the Distance
  6. Height Three Gorenstein Algebras
  7. Proof of Theorem \ref{['thm:IntroMainThm']}
  8. Degree 2d = 10
  9. Case (1):
  10. Case (2):
  11. Case (3):
  12. Degree 2d = 12
  13. Case (1):
  14. Case (2):
  15. Case (3):
  16. ...and 5 more sections

Key Result

theorem 1.1

If $2d = 8$, $2d = 10$, or $2d = 12$, then $\mathop{\rm py}(3,2d) = d+1$.

Figures (1)

  • Figure 1: Partition of a $6\times 4$ block corresponding to the Hilbert function $T = (1,3,6,9,10,9,6,3,1)$ of a Gorenstein ideal with socle degree $8$ and minimal generator degree 3. The table displays the degrees of generators ($q_i$), degrees of relations ($p_i$), diagonal degrees $(r_i)$. Since $r_4 + r_6 = r_5+r_7 = 0$, there is an ideal $J$ generated by a cubic and two quartics which gives rise to this Hilbert function.

Theorems & Definitions (36)

  • theorem 1.1
  • Lemma 2.1: See e.g. Froberg2012WaringScheiderer2017SOSLengthRealForms
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • ...and 26 more