Semialgebraic Splines

We display two C⁰-splines and three C¹-splines defined on the unit circle in R² subdivided into three regions which have curvilinear boundaries. One boundary is along the negative y-axis and the other two are arcs of circles centred at (0,1) and (1,-1), respectively.

The movies of C0 splines were created using the maple file C0_spline.maple. The movies of C1 splines were created using the maple file C1_spline.maple.

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In our paper "Semialgebraic Splines", we have several computations, many with Macaulay 2.

• Table1.m2. This file does the computation for Table 1, which is the Hilbert function of the spline spaces for the cell complex of Figure 1, for r=1..6. Here is its output
• Corollary8.maple. This file does the simplification leading to Corollary 8 of the degree of S/I, when I is generated by t forms of degree n which are pairwise relatively prime. It also generates a table of the Hilbert function. Here is its output when t=3.
• Corollary10.maple. This file does the simplification leading to Corollary 10 of the degree of S/J, when J is generated by members of a pencil of t forms of degree n which are pairwise relatively prime. It also generates a table of the Hilbert function. Here is its output when t=3.
• Pencils of Quadrics. This page is a companion to Remark 11, which illustrated that for a pencil of three quadrics, the dimensions of the spline modules do not depend upon the geometry of the curves underlying the edges, just that they define a real scheme of degree four given by a complete intersection of two conicd.
• SaturationCounterexample.m2. This file does the computation of Remark 11, which shows that the saturation of J with m is not the ideal I, even though J and I have the same multiplicity. Here is its output

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Last modified: Fri May 10 09:01:26 CST 2019