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PreprintarXiv licencearXiv

Is the Signed Zarankiewicz Number the Same as the Recursive-line Zarankiewicz Number?

  • Zhiwei Chen
  • Yannan Chen
  • Liqun Qi
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Abstract

Löfberg and Qi introduced the second order Zarankiewicz number z2, the recursive-line Zarankiewicz number zRL, and the signed Zarankiewicz number zSL for doubly simple biquadratic forms. It was shown that z2(m,n)≥ zSL(m,n)≥ zRL(m,n) for all m and n. However, there was no evidence that there exist particular m and n such that zSL(m,n)>zRL(m,n). The motivation for introducing zSL was as follows: during the study of the exceptional case m=15, n=6, Löfberg and Qi showed that z2(15,6)=zSL(15,6)=60, but the exact value of zRL(15,6) was unknown then. In this paper we show that zRL(15,6)=60. This eliminates the motivation for introducing zSL. Whether zSL(m,n)=zRL(m,n) in general remains an open problem. Recently, Lebedev presented an explicit construction separating the augmented Zarankiewicz number zA from the limited augmented Zarankiewicz number zL at m=n=1893. We hope that the separation problem for zSL and zRL can also be solved. We also present the exact values of zRL(m,6) for 6≤ m≤ 16.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026signed,
  title = {{Is the Signed Zarankiewicz Number the Same as the Recursive-line Zarankiewicz Number?}},
  author = {Zhiwei Chen and Yannan Chen and Liqun Qi},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.20071},
  eprint = {2609.20071},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.20071},
}
RIS
TY  - UNPB
TI  - Is the Signed Zarankiewicz Number the Same as the Recursive-line Zarankiewicz Number?
AU  - Zhiwei Chen
AU  - Yannan Chen
AU  - Liqun Qi
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2609.20071
UR  - https://arxiv.org/abs/2609.20071
AB  - Löfberg and Qi introduced the second order Zarankiewicz number z_2, the recursive-line Zarankiewicz number z_RL, and the signed Zarankiewicz number z_SL for doubly simple biquadratic forms. It was shown that z_2(m,n)≥ z_SL(m,n)≥ z_RL(m,n) for all m and n. However, there was no evidence that there exist particular m and n such that z_SL(m,n)>z_RL(m,n). The motivation for introducing z_SL was as follows: during the study of the exceptional case m=15, n=6, Löfberg and Qi showed that z_2(15,6)=z_SL(15,6)=60, but the exact value of z_RL(15,6) was unknown then. In this paper we show that z_RL(15,6)=60. This eliminates the motivation for introducing z_SL. Whether z_SL(m,n)=z_RL(m,n) in general remains an open problem. Recently, Lebedev presented an explicit construction separating the augmented Zarankiewicz number z_A from the limited augmented Zarankiewicz number z_L at m=n=1893. We hope that the separation problem for z_SL and z_RL can also be solved. We also present the exact values of z_RL(m,6) for 6≤ m≤ 16.
ER  -