On Wednesday, August 5, 2020 at 4:28:42 AM UTC-7, Santanu wrote:
>
> Dear all,
>     Consider ideal I=<x0*x1+x2> over the binary field GF(2). 
> Then (x2).reduce(I) gives x2. I want it to be x0*x1. 
> In fact , I want this kind of reduction always should give quadratic 
> polynomial 
> (I know that this is possible for my problems).
>
> That means you should try and put a monomial order on your ring that gives 
x2 a higher weight than x0*x1. "lex" would do that. It may be that you need 
to order your variables as [x2,x0,x1] instead (i.e., that the first 
variable has the highest weight)

I don't know what you mean by "should always give a quadratic polynomial". 
With your ideal, x2^3 is not reducible to a quadratic polynomial. Do you 
want to work with a Boolean ring instead, where the relations 
x0^2-x0=1^2-x1=x2^2-x2=0 are implied as well?
 

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