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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2507.16399 |
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| _version_ | 1866909930022240256 |
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| author | Qi, Liqun Cui, Chunfeng Xu, Yi |
| author_facet | Qi, Liqun Cui, Chunfeng Xu, Yi |
| contents | In 1973, Calderón proved that an $m \times 2$ positive semidefinite (psd) biquadratic form can always be expressed as the sum of ${3m(m+1) \over 2}$ squares of quadratic forms. Very recently, by applying Hilbert's theorem on ternary quartics, we proved that a $2 \times 2$ psd biquadratic form can always be expressed as the sum of three squares of bilinear forms. This improved Calderón's result for $m=2$, and left the sos (sum-of-squares) rank problem of $m \times 2$ biquadratic forms for $m \ge 3$ to further exploration. In this paper, we show that an $3 \times 2$ psd biquadratic form can always be expressed as four squares of bilinear forms. We make a conjecture that an $m \times 2$ psd biquadratic form can always be expressed as $m+1$ squares of bilinear forms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_16399 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The SOS Rank of Biquadratic Forms Qi, Liqun Cui, Chunfeng Xu, Yi Number Theory In 1973, Calderón proved that an $m \times 2$ positive semidefinite (psd) biquadratic form can always be expressed as the sum of ${3m(m+1) \over 2}$ squares of quadratic forms. Very recently, by applying Hilbert's theorem on ternary quartics, we proved that a $2 \times 2$ psd biquadratic form can always be expressed as the sum of three squares of bilinear forms. This improved Calderón's result for $m=2$, and left the sos (sum-of-squares) rank problem of $m \times 2$ biquadratic forms for $m \ge 3$ to further exploration. In this paper, we show that an $3 \times 2$ psd biquadratic form can always be expressed as four squares of bilinear forms. We make a conjecture that an $m \times 2$ psd biquadratic form can always be expressed as $m+1$ squares of bilinear forms. |
| title | The SOS Rank of Biquadratic Forms |
| topic | Number Theory |
| url | https://arxiv.org/abs/2507.16399 |