Semialgebraic splines

Bivariate semialgebraic splines

“…This was refined by Billera and Rose [3,4] and by Schenck and Stillman [19,20], who viewed splines as a graded module over the polynomial ring, so that the dimension of spline spaces is given by the Hilbert function of the module. In [12] we observed that this homological approach remains valid for semialgebraic splines. We recall that.…”

“…We determine the Hilbert polynomial of C r (∆) for a generic mesh ∆. We explain exactly what we mean by a generic mesh in Definition 4.5, which extends conditions from [12] by an acyclicity condition on ∆. Our arguments follow the those in [19,20] with modifications reflecting the more complicated geometry of the mesh as in [10,17].…”

“…With Sun, we observed that this homological machinery carries over to semialgebraic meshes [12]. We used this to determined the dimensions of spline spaces for two distinct generalizations of rectilinear meshes when there is a single interior vertex.…”

“…We consider two cases of semialgebraic meshes extending those from [12]. For the first, we assume that every edge form lies in the vector space spanned by three forms of degree n (called a net).…”

“…The second case is of splines over generic semialgebraic meshes. In addition to the conditions from [12] for genericity at interior vertices, we also assume a certain acyclicity condition on ∆ that appeared in [17]. These conditions are satisfied for almost all meshes ∆ and thus by generic meshes in the algebraic geometry sense.…”