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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.00775 |
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| _version_ | 1866915977959047168 |
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| author | Barseghyan, Diana Bory-Reyes, Juan Schneider, Baruch Zhang, Yifan |
| author_facet | Barseghyan, Diana Bory-Reyes, Juan Schneider, Baruch Zhang, Yifan |
| contents | We construct an intrinsic q-deformation of the vector derivative on radial algebras. The construction is not obtained from a coordinate realization by replacing ordinary partial derivatives with one-variable Jackson derivatives; that coordinatewise procedure does not preserve radial subalgebras. Instead, for each distinguished vector variable $x$ and each finite set of auxiliary variables $Y\subset S\setminus\{x\}$, we define a q-Cartan derivative $\partial^Y_{x,q}$ on $R(\{x\}\cup Y)$ using the $x$-relative scalar variables $x^2$ and $\{x,y\}$, $y\in Y$. We prove two Fischer-type theorems. First, an exterior Fischer operator has a triangular anticommutator with explicit resonance factors; after inverting them one obtains a global Green operator and an exterior direct-sum decomposition. Second, using full left multiplication by $x$, we prove the monogenic Fischer decomposition after localization by finite-block determinants. We also describe the first denominator factors: the one-vector and two-vector factors are explicit, while the general determinant factors split by $x$-support. A degree-zero support-rank obstruction shows that a universal unlocalized theorem for all real $0<q<1$ cannot hold without excluding q-resonances. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_00775 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Intrinsic \(q\)-Radial Vector Derivatives and Localized Fischer Decompositions on Radial Algebras Barseghyan, Diana Bory-Reyes, Juan Schneider, Baruch Zhang, Yifan Complex Variables We construct an intrinsic q-deformation of the vector derivative on radial algebras. The construction is not obtained from a coordinate realization by replacing ordinary partial derivatives with one-variable Jackson derivatives; that coordinatewise procedure does not preserve radial subalgebras. Instead, for each distinguished vector variable $x$ and each finite set of auxiliary variables $Y\subset S\setminus\{x\}$, we define a q-Cartan derivative $\partial^Y_{x,q}$ on $R(\{x\}\cup Y)$ using the $x$-relative scalar variables $x^2$ and $\{x,y\}$, $y\in Y$. We prove two Fischer-type theorems. First, an exterior Fischer operator has a triangular anticommutator with explicit resonance factors; after inverting them one obtains a global Green operator and an exterior direct-sum decomposition. Second, using full left multiplication by $x$, we prove the monogenic Fischer decomposition after localization by finite-block determinants. We also describe the first denominator factors: the one-vector and two-vector factors are explicit, while the general determinant factors split by $x$-support. A degree-zero support-rank obstruction shows that a universal unlocalized theorem for all real $0<q<1$ cannot hold without excluding q-resonances. |
| title | Intrinsic \(q\)-Radial Vector Derivatives and Localized Fischer Decompositions on Radial Algebras |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2605.00775 |