I don't think that holds for zero polynomials, since a zero polynomial times any other polynomial yields a zero polynomial. I suppose you could go with a zero polynomial having negative infinite degree, to make the sum of polynomial degrees be the degree of their products, but I don't see how that changes a concept of representing polynomials in J:
The degree of a polynomial is less than the length of the list representing it in J, but negative infinity is less than the length of any J list. Thanks, -- Raul On Tue, Jan 3, 2023 at 10:54 PM Elijah Stone <[email protected]> wrote: > > Ah, I see--per wikipedia, it's so that the degree of the product of two > polynomials is the sum of their degrees. Eh. > > On Tue, 3 Jan 2023, Henry Rich wrote: > > > That's the most common definition of polynomial degree. > > > > Henry Rich > > > > On 1/3/2023 7:55 PM, Elijah Stone wrote: > >> A polynomial is a sum. An empty polynomial has no summands; and no > >> summands sum to 0. > >> > >> An empty polynomial should have degree 0. Why do you say a polynomial > >> with a singular term of 0 has degree __? > >> > >> I agree with raul. > >> > >> On Tue, 3 Jan 2023, Henry Rich wrote: > >> > >>> I disagree. A polynomial must have at least one term, in > >>> mathematical usage. A polynomial with only one term is already a > >>> weird case: all except the zero polynomial have no roots; and if the > >>> term is nonzero, the degree of the polynomial is considered 0, but if > >>> the term is 0, the degree is __ . What would the degree of the empty > >>> polynomial be? > >>> > >>> The derivative of a polynomial with 1 constant term should be the > >>> zero polynomial, not empty. > >>> > >>> What would the integral of the empty polynomial be? > >>> > >>> I think the original implementation of p. was wrong. > >>> > >>> Henry Rich > >>> > >>> On 1/3/2023 5:06 PM, Raul Miller wrote: > >>>> In J807: > >>>> > >>>> (i.0) p. 2 > >>>> 0 > >>>> > >>>> In the current j904 beta: > >>>> > >>>> (i.0) p. 2 > >>>> |domain error, executing dyad p. > >>>> |polynomial may not be empty > >>>> | (i.0) p.2 > >>>> > >>>> I think this change should be reverted. > >>>> > >>>> I have not researched the full history of this change, but J > >>>> polynomials may be padded with an arbitrary number of trailing zeros > >>>> without changing their significance. So, mathematically speaking, > >>>> empty polynomials should be within the domain of the p. verb. > >>>> > >>>> But, also, this breaks typical general case implementation of a > >>>> derivative operation on polynomials. In other words, this worked fine > >>>> under j807: > >>>> > >>>> pderiv=: 1 }. (* i.@#) > >>>> (pderiv 5) p. 2 > >>>> 0 > >>>> > >>>> Thanks, > >>>> > >>> > >>> ---------------------------------------------------------------------- > >>> For information about J forums see http://www.jsoftware.com/forums.htm > >>> > >> ---------------------------------------------------------------------- > >> For information about J forums see http://www.jsoftware.com/forums.htm > > > > ---------------------------------------------------------------------- > > For information about J forums see http://www.jsoftware.com/forums.htm > > > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
