"Bill Page" writes : > > > Could you please define in what sense "Otherwise we don't > > > get a polynomial, obviously."? To me this is not obvious - > > > it is wrong. > > > > Why should this be wrong? > > > > Here's a definition for polynomial from wikipedia: > > > > In mathematics, a polynomial is an expression in which constants > > and variables are combined using (only) addition, subtraction, > > and multiplication. Thus, 7x^2+4x-5 is a polynomial; 2/x is not.
I have the same point of view than Martin, everything is done in DMP in order to have a polynomial ring, DMP isn't only a << collect >> command which factorise x and so. > I think this definition is less general it needs to be and > difficult to apply in the context of Axiom. If we change DMP assumption, it's no more ring. > We need to know what are "constants" and what are "variables" So the same variable is forbiden in the constant ring. I find it's a bug to allow DMP ([X,Z], DMP ([X,Y], INT)) > > I recall that in the Algebra course I attended, the polynomial > > ring was defined as a ring (of coefficients) together with a > > variable which is to be transcendent over that ring. > > That definition is much better. In the case we are discusing > the ring of coefficients is the domain 'EXPR INT' which > includes expressions of the form '1/x' and so it is correct > to say that '2/x' is not in itself a polynomial but it can > be a coefficient in a polynomial. I don't think so, a polynomial ring has variables which are not in the initial ring. The DMP error (for derivative) isn't in the derivative function but in the fact that axiom accepts 1/x has variable in DMP[x]. And 1/x isn't transcendent over that rign because y=1/x is solution of xy-1=0 in this field. Either we say to the user he _must_ be carreful with this, (only in the EXPR INTEGER field?) Either monomial command of DMP reject 1/x coefficient if x is a variable. In France we say we can be << jesuite >> : The real axiom accepts coefficients as 1/x in DMP ([x], EXPR INT) because axiom EXPR INT isn't perfect, but it souldn't. But I'm sure it's a mistake to change derivative. > How would you use such a domain to solve the problem originally > posed by Francois about expansion of trigonometric expressions > that started this thread? When we expand a trigonometric expression, the question is a << linear one >> not a polynomial one. We expand more often sin (4*x+5*y) than sin (x^2+x) ; for sin (x^2+x) we use series. Have a good day ! Francois _______________________________________________ Axiom-developer mailing list [email protected] http://lists.nongnu.org/mailman/listinfo/axiom-developer
