Hi,

Look at this.

sage: tord = TermOrder(matrix([3,2,4,1,1,0,1,0,0])) 
sage: S.<t>=PolynomialRing(QQ)
sage: R.<x,y,z> = PolynomialRing(S,'x',3,order=tord)
sage: (x^2).degree()
2

So the behavior is not consistent among different backend engines, namely, 
Singular and PolyDict. I think degree() should return the total degree, not 
the weighted degree, as the current docstring says. So the current behavior 
with Singular engine should be considered as a bug. I think there should be 
a separate method like weighted_degree() that return the proper weighted 
degree, and for that, there should be a mechanism to set weights of the 
variables. I am not sure whether it is always desirable that the weighted 
term order determines the weights of the variables. I think in principle 
they can be independent.

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