Given an ideal in a polynomial ring, we show that the asymptotic resurgence studied by Guardo, Harbourne, and Van Tuyl can be computed using integral closures. As a consequence, the asymptotic resurgence of an ideal is the maximum of finitely many ratios involving Waldschmidt-like constants (which we call skew Waldschmidt constants) defined in terms of Rees valuations. We use this to prove that the asymptotic resurgence coincides with the resurgence if the ideal is normal (that is, all its powers are integrally closed).For a monomial ideal the skew Waldschmidt constants have an interpretation involving the symbolic polyhedron defined by Cooper, Embree, Hà, and Hoefel. Using this intuition we provide several examples of squarefree monomial ideals whose resurgence and asymptotic resurgence are different.
Let M and N be two monomials of the same degree, and let I be the smallest Borel ideal containing M and N . We show that the toric ring of I is Koszul by constructing a quadratic Gröbner basis for the associated toric ideal. Our proofs use the construction of graphs corresponding to fibers of the toric map. As a consequence, we conclude that the Rees algebra is also Koszul.
The dimension of planar splines on polygonal subdivisions of degree at most d is known to be a degree two polynomial for d 0. For planar C r splines on triangulations this formula is due to Alfeld and Schumaker; the formulas for planar splines with varying smoothness conditions across edges on convex polygonal subdvisions are due to Geramita, McDonald, and Schenck. In this paper we give a bound on how large d must be for the known polynomial formulas to give the correct dimension of the spline space. Bounds are given for central polytopal complexes in three dimensions, or polytopal cells, with varying smoothness across two-dimensional faces. In the case of tetrahedral cells with uniform smoothness r we show that the known polynomials give the correct dimension for d ≥ 3r + 2; previously Hong and separately Ibrahim and Schumaker had shown that this bound holds for planar triangulations. All bounds are derived using techniques from computational commutative algebra.
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