Shifted Partial Derivative Polynomial Rank and Codimension
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915692923584512 |
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| author | Edwards, Darren J. |
| author_facet | Edwards, Darren J. |
| contents | Shifted partial derivative (SPD) methods are a central algebraic tool for circuit lower bounds, measuring the dimension of spaces of shifted derivatives of a polynomial. We develop the Shifted Partial Derivative Polynomial (SPDP) framework, packaging SPD into an explicit coefficient-matrix formalism. This turns shifted-derivative spans into concrete linear-algebraic objects and yields two dual measures: SPDP rank and SPDP codimension.
We define the SPDP generating family, its span, and the SPDP matrix M_{k,l}(p) inside a fixed ambient coefficient space determined by the (k,l) regime, so rank is canonical and codimension is a deficit from ambient fullness. We prove structural properties needed for reuse: monotonicity in the shift/derivative parameters (with careful scoping for |S|=k versus |S|<=k conventions), invariance under admissible variable symmetries and basis changes, and robustness across standard Boolean/multilinear embeddings. We then give generic width-to-rank upper-bound templates for local circuit models via profile counting, separating the model-agnostic SPDP toolkit from additional compiled refinements used elsewhere. We illustrate the codimension viewpoint on representative examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_20729 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Shifted Partial Derivative Polynomial Rank and Codimension Edwards, Darren J. Computational Complexity 68Q17 68Q15 13P25 F.1.3; F.2.1 Shifted partial derivative (SPD) methods are a central algebraic tool for circuit lower bounds, measuring the dimension of spaces of shifted derivatives of a polynomial. We develop the Shifted Partial Derivative Polynomial (SPDP) framework, packaging SPD into an explicit coefficient-matrix formalism. This turns shifted-derivative spans into concrete linear-algebraic objects and yields two dual measures: SPDP rank and SPDP codimension. We define the SPDP generating family, its span, and the SPDP matrix M_{k,l}(p) inside a fixed ambient coefficient space determined by the (k,l) regime, so rank is canonical and codimension is a deficit from ambient fullness. We prove structural properties needed for reuse: monotonicity in the shift/derivative parameters (with careful scoping for |S|=k versus |S|<=k conventions), invariance under admissible variable symmetries and basis changes, and robustness across standard Boolean/multilinear embeddings. We then give generic width-to-rank upper-bound templates for local circuit models via profile counting, separating the model-agnostic SPDP toolkit from additional compiled refinements used elsewhere. We illustrate the codimension viewpoint on representative examples. |
| title | Shifted Partial Derivative Polynomial Rank and Codimension |
| topic | Computational Complexity 68Q17 68Q15 13P25 F.1.3; F.2.1 |
| url | https://arxiv.org/abs/2512.20729 |