Executive Summary : | Let (R,m) be a Noetherian local ring with the maximal ideal m and positive dimension d. Let q be an m-primary ideal of R. A broader goal of this project is to solve problems on the following numerical invariant of q: (1) Reduction number of q: An ideal J contained in q is called a reduction of q if q^{n+1}=Jq^n for large values of n. The reduction number of q (RedN(q)) with respect to J is the smallest n such q^{n+1}=Jq^n. (2) The Hilbert coefficients of q: The d+1 unique integers e_0(q),e_1(q),...,e_d(q) such that the length function \lambda(A/q^n) coincides with a rational polynomial of degree d which has the above coefficients when written in a particular way. Both the above invariant are data of the asymptotic powers of q but they capture deep information about the ideal q, the local ring R and the blow-up algebras associated to q. In general, it is hard to compute RedN(q). Finding a computable upper bound on RedN(q) is an important problem. A remarkable bound was proved by M. E. Rossi. Suppose R is a Cohen-Macaulay(CM) local ring of dimension at most two. Rossi proved that RedN(q)is bounded above by e_1(q)-e_0(q)+\lambda(R/q)+1. Proving the same bound in dimension strictly larger than two is an open problem. The problems proposed in this project are attempts to move closer to Rossi's bound in higher dimensions. We propose to find a polynomial f with variables e_0(q),...,e_d(q) and of degree at most two which bounds RedN(q) from above. Our ambition is to find such a polynomial of degree one. The first three Hilbert coefficients of q are investigated well but very less is known about e_i(q) when i is larger than two unless depth of the associated graded ring G(q) of q is high enough. One goal of this project is to solve problems on e_3(q) and e_4(q) such as finding their range and their interaction with RedN(q) and with lower Hilbert coefficients. Further we want to use it to find new bounds for RedN(q) and to find certain homological invariant of G(q) such as the a-invariant, I-invariant, Hilbert series etc. We propose to investigate the invariant e_3(q)-e_2(q)(e_2(q)-1). We believe it is a non-positive number. What are the consequences of its vanishing on the associated graded ring? Further, let q be an integrally closed m-primary ideal. While working with e_3(q), an important invariant involving lower Hilbert coeffcients turns up which is e_2(q)- e_1(q)+ e_0(q)-\lambda(R/q). This number is known to be non-negative for an integrally closed m-primary ideal q. In that case, there are strong evidences that e_3(q) is bounded above by e_2(q)- e_1(q)+ e_0(q)-\lambda(R/q). What are the consequences of the boundary conditions for the above bound of e_3(q) on the homological and numerical invariant of G(q). We also propose to study the equality e_2(q)- e_1(q)+ e_0(q)=\lambda(R/q) in the spirit of extending recent results of Mishra-Puthepurakal, 2022 from the case when q=m to an integrally closed m-primary ideal q. |