Algebraic methods to study the dimension of supersmooth spline spaces

Multivariate piecewise polynomial functions (or splines) on polyhedral complexes have been extensively studied over the past decades and find applications in diverse areas of applied mathematics including numerical analysis, approximation theory, computer aided geometric design. In this paper we address various challenges arising study splines with enhanced mixed (super-)smoothness conditions at vertices across interior faces partition. Such supersmoothness can be imposed but also appear unexpectedly certain depending geometry underlying Using algebraic tools, a generalization Billera–Schenck–Stillman complex that includes effect additional smoothness constraints leads to construction which requires analysis ideals generated by products powers linear forms several variables. Specializing case planar triangulations, combinatorial lower bound dimension is presented, show gives exact high degree. The methods are further illustrated examples.