A dimension series for multivariate splines

DOI10.1007/BF02574678zbMath0725.13011OpenAlexW2136354177MaRDI QIDQ2277525

Lauren L. Rose, Billera, Louis J.

Publication date: 1991

Published in: Discrete \& Computational Geometry (Search for Journal in Brave)

Full work available at URL: https://eudml.org/doc/131147

zbMATH Keywords

Gröbner basisgenerating functionmultivariate splinespiecewise polynomial functions

Mathematics Subject Classification ID

Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) (13P10) Polynomials, factorization in commutative rings (13P05)

Related Items

Algebraic methods in approximation theory ⋮ Splines in geometry and topology ⋮ Piecewise polynomials on polyhedral complexes ⋮ A family of ideals of minimal regularity and the Hilbert series of \(C^r(\widehat\Delta)\) ⋮ A spectral sequence for splines ⋮ From spline approximation to Roth's equation and Schur functors ⋮ Semialgebraic splines ⋮ Multivariate $$C^1$$-Continuous Splines on the Alfeld Split of a Simplex ⋮ Lattice-supported splines on polytopal complexes ⋮ Shellability and freeness of continuous splines ⋮ Sequence decomposition method for computing a Gröbner basis and its application to bivariate spline ⋮ A lower bound for the dimension of tetrahedral splines in large degree ⋮ Dimension of mixed splines on polytopal cells ⋮ Multivariate polynomial splines on generalized oranges ⋮ Generalized splines on graphs with two labels and polynomial splines on cycles ⋮ Multivariate spline and its applications in science and technology ⋮ A homological characterization for freeness of multi-arrangements ⋮ Basis criteria for generalized spline modules via determinant ⋮ The correspondence between multivariate spline ideals and piecewise algebraic varieties ⋮ Splines on the Alfeld split of a simplex and type A root systems ⋮ Hilbert polynomials and geometric lattices ⋮ Splines with boundary conditions ⋮ A new bound for smooth spline spaces ⋮ An algorithm for generating generalized splines on graphs such as complete graphs, complete bipartite graphs and hypercubes ⋮ Bivariate semialgebraic splines ⋮ Generalized splines and graphic arrangements ⋮ Flow-up bases for generalized spline modules on arbitrary graphs ⋮ Modules of piecewise polynomials and their freeness ⋮ An algorithm for constructing a basis for 𝐶^{𝑟}-spline modules over polynomial rings ⋮ Polynomial splines of non-uniform degree on triangulations: combinatorial bounds on the dimension ⋮ Polynomial spline spaces of non-uniform bi-degree on T-meshes: combinatorial bounds on the dimension ⋮ On a conjecture of Rose. ⋮ Associated primes of spline complexes ⋮ Piecewise polynomials with different smoothness degrees on polyhedral complexes ⋮ Graphs admitting only constant splines ⋮ Fat points, inverse systems, and piecewise polynomial functions ⋮ Equivariant Chow cohomology of nonsimplicial toric varieties ⋮ Real zeros of the zero-dimensional parametric piecewise algebraic variety ⋮ Algebraic methods to study the dimension of supersmooth spline spaces ⋮ A Lower Bound for Splines on Tetrahedral Vertex Stars ⋮ Splines on generalized quasi-cross-cut partitions ⋮ Multivariate spline and algebraic geometry ⋮ Generalized integer splines on arbitrary graphs ⋮ Module Basis for Generalized Spline Modules ⋮ Quasi-splines and their moduli

Cites Work

• A formula for the characteristic polynomial of an arrangement
• Gröbner bases and invariant theory
• The algebra of continuous piecewise polynomials
• The dimension of bivariate spline spaces of smoothness r for degree \(d\geq 4r+1\)
• Hilbert polynomials and geometric lattices
• Combinatorial construction of logarithmic differential forms
• On Smooth Multivariate Spline Functions
• Certain Reflexive Sheaves on P n c and a Problem in Approximation Theory
• Multivariate Perpendicular Interpolation
• An Explicit Basis for $C^1 $ Quartic Bivariate Splines
• Vector Bundles on Complex Projective Spaces and Systems of Partial Differential Equations. I
• Homology of Smooth Splines: Generic Triangulations and a Conjecture of Strang
• Combinatorics and commutative algebra
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item