Open questions of McDonald and Schenck on spline modules over polyhedral complexes: remaining Hilbert-polynomial coefficients, Castelnuovo–Mumford regularity, mixed smoothness, and freeness descent

For a hereditary d-dimensional polyhedral complex $P$, the dimension of the space $C^r_k(P)$ of splines (piecewise polynomial functions of smoothness $r$ and degree at most $k$) is given, for all sufficiently large $k$, by a Hilbert polynomial $f(P,r,k)$ of degree $d$. McDonald and Schenck (2009) computed the first three coefficients of $f$ and closed their paper with four open questions and directions that together ask for the spline module $C^r(\hat P)$ on a polyhedral complex to be understood as completely as it is for planar simplicial complexes: (i) determine the next coefficient, the $k^{d-3}$ term — open even when $P$ is simplicial; (ii) determine the Castelnuovo–Mumford regularity of the sheaf associated to $C^r(\hat P)$, i.e. the sharp degree threshold from which the Hilbert polynomial gives the exact dimension; (iii) extend the formulas to mixed smoothness across faces; and (iv) decide whether freeness of $C^r(\hat P)$ implies freeness of $C^{r-1}(\hat P)$, known only in the planar simplicial case. All four components remain open in the stated generality, with substantial partial progress on the mixed-smoothness and regularity components in the planar and special trivariate settings.

Background

A polyhedral complex $P \subset \mathbb{R}^d$ is a finite collection of convex polytopes (the cells) closed under taking faces, in which any two cells meet in a common face; it is called hereditary if for every face the graph whose vertices are the $d$-cells containing that face and whose edges join pairs sharing a $(d-1)$-cell is connected — the natural polyhedral analog of a subdivision of a manifold. A spline of smoothness $r$ and degree at most $k$ on $P$ is a real-valued function that restricts to a polynomial of degree at most $k$ on each $d$-cell and whose polynomial pieces agree, together with all derivatives up to order $r$, across every shared $(d-1)$-face. Writing $C^r_k(P)$ for the vector space of such splines, computing $\dim_{\mathbb{R}} C^r_k(P)$ is the fundamental dimension problem of spline theory: it governs the construction of bases for the function spaces underlying finite element analysis and geometric modeling, and it remains open outside planar simplicial meshes.

The algebraic approach, initiated by Billera (1988) in his resolution of Strang's conjecture on generic planar $C^1$ splines, organizes all degrees at once into a module $C^r(\hat P)$ over the polynomial ring $R=\mathbb{R}[x_1,\dots,x_{d+1}]$ in $d+1$ variables, where $\hat P$ is the homogeneous cone over $P$, so that $C^r_k(P)$ is the degree-$k$ graded piece. Standard commutative algebra then implies that $\dim C^r_k(P)$ agrees, for all sufficiently large $k$, with a polynomial $f(P,r,k)$ of degree $d$, the Hilbert polynomial of the module. For a planar simplicial complex, Alfeld and Schumaker give an explicit formula for all three coefficients of the quadratic $f$, valid for arbitrary triangulations when $k \ge 4r+1$ (1987) and for generic triangulations down to the borderline degree $k = 3r+1$ (1990); a spectral sequence of Schenck (1997) — a staged homological computation that recovers the homology of the spline module from local data at the faces — shows that in every dimension the three leading coefficients are governed by a higher-dimensional analog of this formula. For general polyhedral complexes, Billera and Rose (1991) obtained the first two coefficients, and McDonald and Schenck (2009) obtained the third (the $k^{d-2}$ term) using localization at codimension-2 linear spaces and a refined dual graph, completing the quadratic when $d=2$.

Two further invariants measure how much remains unknown. First, the Castelnuovo–Mumford regularity of a graded module — the largest shift of $R$ occurring in the tail of a minimal free resolution, minus its position — is, as observed by Schenck and Stiller (2002), exactly the threshold $k_0$ beyond which $\dim C^r_k(P)$ equals its Hilbert polynomial; only bounds on $k_0$ are known in the polyhedral setting. Second, the module $C^r(\hat P)$ may be free, i.e. isomorphic to a direct sum of shifted copies of $R$; freeness is the strongest structural property available, since for a free module the Hilbert series — hence all dimensions — is determined by the ranks and shifts in a resolution (Billera–Rose 1992), and Yuzvinsky (1992) initiated the study of freeness for polyhedral complexes. For simplicial complexes, freeness is characterized by the vanishing of homology of certain local modules, and Dalbec and Schenck (2001), in a paper settling a conjecture of Rose, showed that for planar simplicial complexes freeness descends: if $C^r(\hat\Delta)$ is free then so is $C^{r-1}(\hat\Delta)$.

Problem Statement

The source paper closes with four open questions and directions, posed in its "Concluding Remarks and Questions", which together ask that the spline module $C^r(\hat P)$ of a hereditary polyhedral complex be understood as completely as it is in the planar simplicial case. The four independent open problems are:

(a) Hilbert polynomial. Determine the coefficient of $k^{d-3}$ in the Hilbert polynomial $f(P,r,k)$ — the fourth term — in terms of explicit combinatorial and local geometric data of $P$. The problem is open even when $P$ is a simplicial complex of dimension $d \ge 3$; no general formula is known for any coefficient beyond the first three.

(b) Regularity. Determine the Castelnuovo–Mumford regularity of the sheaf associated to $C^r(\hat P)$; equivalently, determine the sharp threshold $k_0(P,r)$ such that $\dim C^r_k(P) = f(P,r,k)$ exactly for all $k \ge k_0$.

(c) Mixed smoothness. Extend the Hilbert-polynomial coefficient formulas to a smoothness distribution, i.e. a different prescribed order of smoothness across each pair of adjacent cells of $P$, as Geramita–Schenck did for planar simplicial complexes. Here the coefficient formulas are the ones already known for uniform smoothness (the first three coefficients); this extension does not include the $k^{d-3}$ coefficient of (a).

(d) Freeness descent. Decide whether freeness of $C^r(\hat P)$ implies freeness of $C^{r-1}(\hat P)$ when $P$ is a polyhedral complex; this implication is known only for planar simplicial complexes (Dalbec–Schenck).

The common scientific objective is an exact and effective description of the dimension of $C^r_k(P)$ in every degree, together with the module structure of $C^r(\hat P)$, for general hereditary polyhedral complexes — with no restriction on the dimension $d$, the smoothness $r$, or the shapes of the cells.

References

1. Terry McDonald and Hal Schenck, "Piecewise polynomials on polyhedral complexes", Advances in Applied Mathematics 42 (2009) 82–93. DOI: 10.1016/j.aam.2008.06.001 (arXiv:0808.2031). https://doi.org/10.1016/j.aam.2008.06.001
2. Peter Alfeld and Larry L. Schumaker, "The dimension of bivariate spline spaces of smoothness r for degree d ≥ 4r+1", Constructive Approximation 3 (1987) 189–197. DOI: 10.1007/BF01890563. https://doi.org/10.1007/BF01890563
3. Peter Alfeld and Larry L. Schumaker, "On the dimension of bivariate spline spaces of smoothness r and degree d = 3r+1", Numerische Mathematik 57 (1990) 651–661. DOI: 10.1007/BF01386434. https://doi.org/10.1007/BF01386434
4. Louis J. Billera, "Homology of smooth splines: generic triangulations and a conjecture of Strang", Transactions of the American Mathematical Society 310 (1988) 325–340. DOI: 10.1090/S0002-9947-1988-0965757-9. https://doi.org/10.1090/S0002-9947-1988-0965757-9
5. Louis J. Billera and Lauren L. Rose, "A dimension series for multivariate splines", Discrete & Computational Geometry 6 (1991) 107–128. DOI: 10.1007/BF02574678. https://doi.org/10.1007/BF02574678
6. Louis J. Billera and Lauren L. Rose, "Modules of piecewise polynomials and their freeness", Mathematische Zeitschrift 209 (1992) 485–497. DOI: 10.1007/BF02570848. https://doi.org/10.1007/BF02570848
7. Sergey Yuzvinsky, "Modules of splines on polyhedral complexes", Mathematische Zeitschrift 210 (1992) 245–254. DOI: 10.1007/BF02571795. https://doi.org/10.1007/BF02571795
8. Hal Schenck, "A spectral sequence for splines", Advances in Applied Mathematics 19 (1997) 183–199. DOI: 10.1006/aama.1997.0534. https://doi.org/10.1006/aama.1997.0534
9. Anthony V. Geramita and Hal Schenck, "Fat points, inverse systems, and piecewise polynomial functions", Journal of Algebra 204 (1998) 116–128. DOI: 10.1006/jabr.1997.7361. https://doi.org/10.1006/jabr.1997.7361
10. John P. Dalbec and Hal Schenck, "On a conjecture of Rose", Journal of Pure and Applied Algebra 165 (2001) 151–154. DOI: 10.1016/S0022-4049(00)00175-4. https://doi.org/10.1016/S0022-4049(00)00175-4
11. Henry K. Schenck and Peter F. Stiller, "Cohomology vanishing and a problem in approximation theory", Manuscripta Mathematica 107 (2002) 43–58. DOI: 10.1007/s002290100222. https://doi.org/10.1007/s002290100222
12. Hal Schenck, "Equivariant Chow cohomology of nonsimplicial toric varieties", Transactions of the American Mathematical Society 364 (2012) 4041–4051. DOI: 10.1090/S0002-9947-2012-05409-2. https://doi.org/10.1090/S0002-9947-2012-05409-2
13. Michael DiPasquale, "Shellability and freeness of continuous splines", Journal of Pure and Applied Algebra 216 (2012) 2519–2523. DOI: 10.1016/j.jpaa.2012.03.010. https://doi.org/10.1016/j.jpaa.2012.03.010
14. Michael DiPasquale, "Lattice-supported splines on polytopal complexes", Advances in Applied Mathematics 55 (2014) 1–21. DOI: 10.1016/j.aam.2013.12.002 (arXiv:1312.3294). https://doi.org/10.1016/j.aam.2013.12.002
15. Michael DiPasquale, "Regularity of mixed spline spaces", preprint, arXiv:1411.2176 (2014). https://arxiv.org/abs/1411.2176
16. Michael DiPasquale, "Dimension of mixed splines on polytopal cells", Mathematics of Computation 87 (2018) 905–939. DOI: 10.1090/mcom/3224. https://doi.org/10.1090/mcom/3224
17. Selma Altınok and Neslihan Ös Sipahi, "Piecewise polynomials with different smoothness degrees on polyhedral complexes", Quaestiones Mathematicae 42 (2019) 673–685. DOI: 10.2989/16073606.2018.1481464. https://doi.org/10.2989/16073606.2018.1481464
18. Beihui Yuan and Michael Stillman, "A counter-example to the Schenck–Stiller '2r+1' conjecture", Advances in Applied Mathematics 110 (2019) 33–41. DOI: 10.1016/j.aam.2019.04.004. https://doi.org/10.1016/j.aam.2019.04.004
19. Deepesh Toshniwal and Michael DiPasquale, "Counting the dimension of splines of mixed smoothness", Advances in Computational Mathematics 47 (2021), article 6. DOI: 10.1007/s10444-020-09830-x (arXiv:2001.01774). https://doi.org/10.1007/s10444-020-09830-x
20. Michael DiPasquale and Nelly Villamizar, "A lower bound for splines on tetrahedral vertex stars", SIAM Journal on Applied Algebra and Geometry 5 (2021) 250–277. DOI: 10.1137/20M1341118. https://doi.org/10.1137/20M1341118
21. Hal Schenck, "Algebraic methods in approximation theory", Computer Aided Geometric Design 45 (2016) 14–31. DOI: 10.1016/j.cagd.2015.11.001 (arXiv:1610.05181). https://doi.org/10.1016/j.cagd.2015.11.001
22. Michael DiPasquale and Nelly Villamizar, "A lower bound for the dimension of tetrahedral splines in large degree", Constructive Approximation 59 (2024) 1–30. DOI: 10.1007/s00365-023-09625-5 (arXiv:2007.12274). https://doi.org/10.1007/s00365-023-09625-5
23. Martina Lanini, Hal Schenck, and Julianna Tymoczko, "The algebra of splines: duality, group actions and homology", preprint, arXiv:2312.09829 (2023). https://arxiv.org/abs/2312.09829
24. Simcha Gilbert, Julianna Tymoczko, and Shira Viel, "Generalized splines on arbitrary graphs", Pacific Journal of Mathematics 281 (2016) 333–364. DOI: 10.2140/pjm.2016.281.333 (arXiv:1306.0801, whose preprint version lists the middle author under the earlier name Shira Polster). https://doi.org/10.2140/pjm.2016.281.333
25. Carles Checa, Michael DiPasquale, Pablo Mazón, Thái Thành Nguyễn, Liana Sega, Prajwal Udanshive, Adam Van Tuyl, and Nelly Villamizar, "Trivariate splines on fans of hyperplane arrangements and Koszul homology", preprint, arXiv:2606.18298 (2026). https://arxiv.org/abs/2606.18298