In this paper we study the class of power ideals generated by the kn forms (x0+ξg1x1+. .+ξgnxn)(k-1)d where ξ is a fixed primitive kth-root of unity and 0≤gj≤k-1 for all j. For k=2, by using a Zkn+1-grading on C[x0,. .,xn], we compute the Hilbert series of the associated quotient rings via a simple numerical algorithm. We also conjecture the extension for k>2. Via Macaulay duality, those power ideals are related to schemes of fat points with support on the kn points [1:ξgjavax.xml.bind.JAXBElement@40ba04c3:. .:ξgjavax.xml.bind.JAXBElement@2e071a4c] in Pn. We compute Hilbert series, Betti numbers and Gröbner basis for these 0-dimensional schemes. This explicitly determines the Hilbert series of the power ideal for all k: that this agrees with our conjecture for k>2 is supported by several computer experiments.
On a class of power ideals / Backelin, J.; Oneto, A.. - In: JOURNAL OF PURE AND APPLIED ALGEBRA. - ISSN 0022-4049. - 219:8(2015), pp. 3158-3180. [10.1016/j.jpaa.2014.10.007]
On a class of power ideals
Oneto A.
2015-01-01
Abstract
In this paper we study the class of power ideals generated by the kn forms (x0+ξg1x1+. .+ξgnxn)(k-1)d where ξ is a fixed primitive kth-root of unity and 0≤gj≤k-1 for all j. For k=2, by using a Zkn+1-grading on C[x0,. .,xn], we compute the Hilbert series of the associated quotient rings via a simple numerical algorithm. We also conjecture the extension for k>2. Via Macaulay duality, those power ideals are related to schemes of fat points with support on the kn points [1:ξgjavax.xml.bind.JAXBElement@40ba04c3:. .:ξgjavax.xml.bind.JAXBElement@2e071a4c] in Pn. We compute Hilbert series, Betti numbers and Gröbner basis for these 0-dimensional schemes. This explicitly determines the Hilbert series of the power ideal for all k: that this agrees with our conjecture for k>2 is supported by several computer experiments.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione
