Polynomial Interpolation of a Vector Field on a Convex Polyhedral Domain
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914301021782016 |
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| author | Chu, Junyan Kaji, Shizuo |
| author_facet | Chu, Junyan Kaji, Shizuo |
| contents | We present a computational method for reconstructing a vector field on a convex polytope $\mathcal{P} \subset \mathbb{R}^d$ of arbitrary dimension from discrete samples. We specifically address the scenario where the vector field is subject to a no-penetration (slip) boundary condition, requiring it to be tangent to the boundary $\partial \mathcal{P}$. Given a degree bound $k$, our algorithm computes a polynomial vector field of degree at most $k$ that fits the observed data in the least-squares sense while exactly satisfying the tangency constraints. Central to our approach is an explicit characterization of the module of polynomial vector fields tangent to $\partial \mathcal{P}$, derived using algebraic concepts from the theory of hyperplane arrangements. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_01803 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Polynomial Interpolation of a Vector Field on a Convex Polyhedral Domain Chu, Junyan Kaji, Shizuo Dynamical Systems Commutative Algebra 37M10, 65D15, 52C35, 41A10, 13N15 We present a computational method for reconstructing a vector field on a convex polytope $\mathcal{P} \subset \mathbb{R}^d$ of arbitrary dimension from discrete samples. We specifically address the scenario where the vector field is subject to a no-penetration (slip) boundary condition, requiring it to be tangent to the boundary $\partial \mathcal{P}$. Given a degree bound $k$, our algorithm computes a polynomial vector field of degree at most $k$ that fits the observed data in the least-squares sense while exactly satisfying the tangency constraints. Central to our approach is an explicit characterization of the module of polynomial vector fields tangent to $\partial \mathcal{P}$, derived using algebraic concepts from the theory of hyperplane arrangements. |
| title | Polynomial Interpolation of a Vector Field on a Convex Polyhedral Domain |
| topic | Dynamical Systems Commutative Algebra 37M10, 65D15, 52C35, 41A10, 13N15 |
| url | https://arxiv.org/abs/2602.01803 |