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Narrowing the Gap: SOS Ranks of $4 \times 3$ Biquadratic Forms and a Lower Bound of $8$

Yi Xu, Chunfeng Cui, Liqun Qi

Abstract

We investigate the maximum sum-of-squares (SOS) rank of biquadratic forms in the critical case of $4 \times 3$ variables, where the general bounds are currently $7 \leq \mathrm{BSR}(4,3) \leq 11$. By analyzing two important structured subclasses, we obtain exact determinations and improved upper bounds that significantly narrow this gap. For simple biquadratic forms those containing only distinct terms of the type $x_i^2 y_j^2$ we prove that the maximum achievable SOS rank is exactly 7, a value attained by a form corresponding to a $C_4$-free bipartite graph with the maximum number of edges. This settles the question for simple forms. For $y$-deficient biquadratic forms a class introduced here that permits cross terms among two of the three $y$-variables while the third appears only in pure square terms we prove an upper bound of $9$ by combining Calderön's theorem on $m\times 2$ forms with the known value $\mathrm{BSR}(4,2) = 5$. Our main result is a constructive proof that $\mathrm{BSR}(4,3) \geq 8$. We present an explicit non-simple, non-deficient $4\times 3$ biquadratic form and prove it requires exactly eight squares, thereby improving the general lower bound. This shows that any form achieving a rank higher than $8$ must possess a more complex algebraic structure, and it reduces the search space for determining the true value of $\mathrm{BSR}(4,3)$. Connections to Zarankiewicz numbers, extremal graph theory, and classical results on sums of squares are highlighted throughout.

Narrowing the Gap: SOS Ranks of $4 \times 3$ Biquadratic Forms and a Lower Bound of $8$

Abstract

We investigate the maximum sum-of-squares (SOS) rank of biquadratic forms in the critical case of variables, where the general bounds are currently . By analyzing two important structured subclasses, we obtain exact determinations and improved upper bounds that significantly narrow this gap. For simple biquadratic forms those containing only distinct terms of the type we prove that the maximum achievable SOS rank is exactly 7, a value attained by a form corresponding to a -free bipartite graph with the maximum number of edges. This settles the question for simple forms. For -deficient biquadratic forms a class introduced here that permits cross terms among two of the three -variables while the third appears only in pure square terms we prove an upper bound of by combining Calderön's theorem on forms with the known value . Our main result is a constructive proof that . We present an explicit non-simple, non-deficient biquadratic form and prove it requires exactly eight squares, thereby improving the general lower bound. This shows that any form achieving a rank higher than must possess a more complex algebraic structure, and it reduces the search space for determining the true value of . Connections to Zarankiewicz numbers, extremal graph theory, and classical results on sums of squares are highlighted throughout.
Paper Structure (6 sections, 9 theorems, 25 equations)

This paper contains 6 sections, 9 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Simple Biquadratic Forms
  3. $y$-Deficient Biquadratic Forms
  4. An Example Illustrating the Bound
  5. An Eight-Square Form and a New Lower Bound for $\operatorname{BSR}(4,3)$
  6. Conclusion

Key Result

Theorem 2.1

The form satisfies $\operatorname{sos}(P_{4,3,7}) = z(4,3) = 7$.

Theorems & Definitions (23)

  • Theorem 2.1: A $4 \times 3$ simple form requiring seven squares
  • Lemma 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4: Hurwitz-type decomposition for $3\times3$ all-ones form
  • proof
  • Proposition 2.5
  • proof
  • Theorem 2.6
  • proof
  • ...and 13 more