Re: [Maxima-discuss] poly_discriminant

[Richard F. <fa...@gm...>]

I think that Michel is right, but maybe there is a related simpler 
explanation...
Perhaps this boils down to whether one is computing over polynomials 
with integer coefficients
or rational number coefficients.  For many users of Maxima,
this is simply not something they think about (Rings, Fields, etc).  For 
others, such considerations
are central to their interests.

consider
gcd(x^2-1,x+1)

   which most people with an opinion would expect to return  x+1.

what about
gcd (x^2/2-1/2,  x+1) ?

maxima returns   (x+1)/2.

How it computes:  by removing the "content"   of each argument, namely 
1/2,  1,  and computing
the gcd of the "primitive part" of the polynomials separately. Then 
sticking 1/2 back on.

But if we are talking about computing over rationals, and are seeking 
the "greatest" common divisor,
which is "greater"?
x+1
   or
(x+1)/2 ?
    and for that matter, there are other common divisors, such as 2*x+2, 
since   1/2*(2*x+2) is x+1, etc.
Which is the greatest?  The route taken by Maxima is to basically 
convert these questions to questions of
polynomials over the integers, where the results are pretty much 
undisputed, and then adjust somehow
in recognition of the possibility that the user had  " something else"  
in mind.

What, for instance, should gcd() return when it is given an argument 
that is clearly not a polynomial?
What do you expect is the most useful return value for ---

gcd(1/x,x+1)  ?

Try it and see if Maxima gives the result you think it should.

If I am not mistaken, the same thinking affects resultant(). Whether it 
should be adjusted or not by
removing contents.   If I am mistaken, I hope you enjoyed the short 
essay on gcd().

RJF

On 6/9/21 7:40 AM, Michel Talon wrote:
>
> Le 09/06/2021 à 16:09, Michel Talon a écrit :
>>
>> At what point has the polynomial been normalized i don't know.
>
> In fact it is in  (rform (rform poly)).  It happens that rform invokes
>
> ratf which gives the "rational" representation from the "general" 
> representation and here:
>
> (ratf poly) returns:
>
> ((MRAT SIMP ($X) (#:X763)) (#:X763 2 1 1 -2 0 6) . 2)
>
> For rationals  fractions, P/Q are represented as dotted pair (P.Q)  
> where polynomials P and Q are
>
> as described previously.  It is clear that taking the rational 
> representation has reduced  coefficients
>
> to a common denominator. You can see this straightforwardly in the 
> maxima command:
>
> (%i7) rat(x^2/2-x+3);
>                                   2
>                                  x  - 2 x + 6
> (%o7)/R/                         ------------
>                                       2
>
> This is the real origin of the discrepancy you have found in the 
> computation of the discriminant to what you
>
> expect.
>
>
>