# multiReesIdeal -- Compute the defining ideal of multi-Rees algebra of ideals

## Synopsis

• Usage:
multiReesIdeal W
• Inputs:
• W, a list, list of ideals $I_1,...,I_n$ over a polynomial ring.
• Optional inputs:
• BasisElementLimit => ..., default value infinity, Bound the number of Groebner basis elements to compute in the saturation step
• DegreeLimit => ..., default value {}, Bound the degrees considered in the saturation step.
• MinimalGenerators => ..., default value true, Whether the saturation step returns minimal generators
• PairLimit => ..., default value infinity, Bound the number of s-pairs considered in the saturation step
• Strategy => ..., default value null, Choose a strategy for the saturation step
• VariableBaseName => ..., default value X, Choose a base name for variables in the created ring
• Outputs:
• an ideal, defining ideal of the multi-Rees algebra of $I_1,...,I_n$.

## Description

The function computes the defining ideal of the multi-Rees algebra of a set of ideals over a polynomial ring by computing the saturation of a binomial ideal with respect to a polynomial. The technique is a generalization of a result of D. Cox, K.-i. Lin and G. Sosa for monomial ideals over a polynomial ring.

 i1 : S = QQ[x_0..x_3] o1 = S o1 : PolynomialRing i2 : C = trim monomialCurveIdeal(S,{2,3,5}) 3 2 3 2 o2 = ideal (x x - x x , x - x x , x - x x ) 1 2 0 3 2 1 3 1 0 2 o2 : Ideal of S i3 : multiReesIdeal {C} 2 2 3 2 3 o3 = ideal (x X - x X - x X , x X - x X - x X , (x - x x )X + (- x + 2 1 1 2 3 3 1 1 0 2 2 3 1 0 2 2 2 ------------------------------------------------------------------------ 2 x x )X ) 1 3 3 o3 : Ideal of S[X ..X ] 1 3 i4 : multiReesIdeal {C,C} 2 2 o4 = ideal (X X - X X , X X - X X , X X - X X , x X - x X - x X , x X - 3 5 2 6 3 4 1 6 2 4 1 5 2 4 1 5 3 6 1 4 ------------------------------------------------------------------------ 2 2 3 2 x X - x X , x X - x X - x X , x X - x X - x X , (x - x x )X + (- 0 5 2 6 2 1 1 2 3 3 1 1 0 2 2 3 1 0 2 5 ------------------------------------------------------------------------ 3 2 3 2 3 2 x + x x )X , (x - x x )X + (- x + x x )X ) 2 1 3 6 1 0 2 2 2 1 3 3 o4 : Ideal of S[X ..X ] 1 6

## Ways to use multiReesIdeal :

• multiReesIdeal(List) (missing documentation)

## For the programmer

The object multiReesIdeal is .