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findElementOfDegree -- find an element of a specified degree

Synopsis

Description

Given a singly graded ring and an integer $n$, this function tries to find an element of degree $n$. If successful, it returns a list with two elements {$a,b$} such that $a/b$ has degree $n$. If it is impossible, it gives an error. If instead of an integer, you pass it a basic list corresponding to a multi-degree, it still tries to find $a, b$ in R such that $a/b$ has the provided multidegree. It only works on rings with flattened variables (ie, no Rees algebras). First we do an example without multidegrees.

i1 : R = ZZ/7[x,y,Degrees=>{3, 5}];
 
i2 : output = findElementOfDegree(1, R)

       2
o2 = {x , y}

o2 : List
 
i3 : output#0/output#1

      2
     x
o3 = --
      y

o3 : frac R
 
i4 : findElementOfDegree(-2, R)

       2   4
o4 = {y , x }

o4 : List

We also do an example with multidegrees

i5 : R = QQ[x,y,Degrees=>{{1,2}, {3, 5}}];
 
i6 : output = findElementOfDegree({1, 3}, R)

       4
o6 = {x , y}

o6 : List
 
i7 : output#0/output#1

      4
     x
o7 = --
      y

o7 : frac R

See also

Ways to use findElementOfDegree :

For the programmer

The object findElementOfDegree is a method function.