The degree of a product of (nonzero) polynomials is the sum of the degrees. If this rule should also apply to the zero polynomial then its degree X should satisfy X+1=X. (Because 0*x=0). No number X satisfies X+1=X.
Sendt fra Yahoo Mail på Android På ons., den 4. jan. 2023 klokken 12:27, Martin Kreuzer<[email protected]> skrev: correction: In last line -unlimited should read negative infinity. -M At 2023-01-04 10:27, you wrote: >Not trying to make a point here, simply citing >E.J. Barbeau, "Polynomials", pp 1,2: >[my transcription- beware of typos] > >\" >In the title of this section, we promised you some anatomy. Here it is. >For the polynomial, a[n] * t^n+ a[n-1] * >t^(n-1)+ ... a[1] * t+ a[0], with a[n] /= 0, the >numbers a[i] (0 <= i <= n) are called oefficients. >a[n] is the leading coefficient, and a[n] * t^n >the leading term, a[0] is the constant term or the constant coefficient. >a[1] is the linear coefficient and a[1] * t the linear term. >When the leading coefficient an is 1, the polynomial is said to be monic. >The nonnegative integer n is the degree of the polynomial; >we write degp = n. >A constant polynomial has but a single term, a[0]. >A nonzero constant polynomial has degree 0, but, >by convention, the zero polynomial (all >coefficients vanishing) has degree -unlimited (__). <<<<<< >"\ > >Thanks. > >-M > >At 2023-01-04 03:40, you wrote: > >>You are essentially saying that ((i. 0) p. y) >>should be defined the same as (0 p. y). I see >>no justification for this. We don't treat (i. >>0) as identical to (0). (i. 0) is not a >>polynomial and trying to make it one is a >>kludge that will come to grief. I think it already has: > >>What is the integral of (i. 0)? > >>If (i. 0) as a polynomial is the same as 0, the >>answer has to be 0 1, which is inconsistent with simple definitions. > >>It is not always a good thing to expand the >>domain of a verb. If you expanded the domain >>of + to include characters, there would be >>fewer errors but much more gnashing of teeth by >>users. I say that if a user thinks that x is a >>polynomial and x is empty, they have a bug and >>the sooner they learn that, the better. > >>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
