(matrix{{x}})^2 != matrix{{x^2}}

[pzinn]

As discussed on zulip, right now we have

i1 : R=QQ[x]

o1 = R

o1 : PolynomialRing

i2 : (matrix{{x}})^2 == matrix{{x^2}}

o2 = false

which is borderline absurd.

I've always found this way of dealing with degrees incorrect. If we decide that m=matrix{{x}} is a map (of degree 0) from R^{-1} to R^{0}, which is the current convention, then we should be consistent and also apply it to products and powers, i.e., m^2 should be a map (also of degree 0) from R^{-2} to R^{0}.

I believe the culprit is this section of Matrix * Matrix in matrix.m2:

	  dif := degrees P - degrees Q;
	  deg := (
	       if #dif === 0
	       then degree m + degree n
	       else if same dif
	       then degree m + degree n + dif#0
 	       else toList (degreeLength ring m:0)
	       );

the degree of m*n should always be degree m+degree n (in particular, preserving the degree 0 case, i.e., preserving homogeneity of maps). So instead of the code above, one should degree-shift the source and target of n to align the target of n and the source of m.

Incidentally this would also resolve another annoying issue:

i3 : (matrix{{x}})^0
stdio:3:13:(3): error: expected source and target to agree