Degree of zero element

[sebastianpos]

There appears to be no consistent behaviour when we ask for the degree of the zero element in a graded ring. Here are some examples:

gap> S1 := GradedRing( HomalgFieldOfRationalsInSingular()*"x" );;
gap> SetWeightsOfIndeterminates( S1, [ 1 ] );;
gap> Degree( Zero( S1 ) );
-1

gap> S2 := GradedRing( HomalgFieldOfRationalsInSingular()*"x" );;
gap> SetWeightsOfIndeterminates( S2, [ -1 ] );;
gap> Degree( Zero( S2 ) );
1

gap> S3 := GradedRing( HomalgFieldOfRationalsInSingular()*"x,y" );;
gap> SetWeightsOfIndeterminates( S3, [ 1,-1 ] );;
gap> Degree( Zero( S3 ) );
-1

gap> S4 := GradedRing( HomalgFieldOfRationalsInSingular()*"x,y" );;
gap> SetWeightsOfIndeterminates( S4, [ [1,0],[0,1] ] );;
gap> Degree( Zero( S4 ) );
( -1, -1 )

gap> S5 := GradedRing( HomalgFieldOfRationalsInSingular()*"x,y" );;
gap> SetWeightsOfIndeterminates( S5, [ [-1,-1],[0,1] ] );;
gap> Degree( Zero( S5 ) );
( -1, -1 )

gap> S5 := GradedRing( HomalgFieldOfRationalsInSingular()*"x,y" );;
gap> SetWeightsOfIndeterminates( S5, [ [-1,0],[0,-1] ] );;
gap> Degree( Zero( S5 ) );
( -1, -1 )

or

gap> R := GradedRing( HomalgFieldOfRationalsInSingular()*"x,y,z" );;
gap> SetWeightsOfIndeterminates( R, [ 1,1,1 ] );;
gap> S := R/"x*y-z";
Q[x,y,z]/( x*y-z )
(weights: yet unset)
gap> WeightsOfIndeterminates( S );
[ 1, 1, 1 ]
gap> Degree( Zero( S ) );
-1
gap> Degree( "x*y-z"/S );
2

Is this intentional?

Mathematically, I would expect either an error when calling Degree of the zero element, or an extra symbol as an output like fail.

[mohamed-barakat]

Back then I wanted to use something like -infinity which was introduced later to GAP:

gap> -infinity + 1000;
-infinity

But since we are now using multi-degrees we cannot use -infinity:

gap> -infinity + [1000,1000];
[ -infinity, -infinity ]

The option to use [ -infinity, -infinity ] is also problematic, since a (multi-)degree is a an element of an IsHomalgModule over HomalgRingOfIntegers().

The clean solution is to introduce a new symbol -inf and overload the + operation to behave as expected with homalg's degrees and multi-degrees.

One can also consider the overloading of + for -infinity and homalg's (multi-)degrees.

[fingolfin]

If the multi degrees are custom objects, and in particular not lists, then I think it makes a lot of sense to overload addition of such a multidegree with -infinity. It should only require a 2-3 lines of code, too.