Re: [Maxima-discuss] poly_discriminant

[Michel T. <ta...@lp...>]

Le 08/06/2021 à 22:57, Karen Kharatian a écrit :
> poly_discriminant(a*x^2+b*x+c, x) → b^2-4*a*c (correct)
>
> but
>
> poly_discriminant(x^2/2-x+3, x) → -20 (incorrect)
> the right answer is ∆ = -5
>
> So, Maxima's poly_discriminant and resultant functions work 
> incorrectly (see my previous message: 
> https://sourceforge.net/p/maxima/mailman/message/37298869/ 
> <https://sourceforge.net/p/maxima/mailman/message/37298869/>)
>
>
I disagree.  If you compute

(%i1) poly_discriminant(x^2 - 2*x+6, x);
(%o1)                                - 20

So you get what you call the correct result when your polynomial has 
been multiplied by 2, that is normalized. Indeed if you trace what

maxima does when computing the discriminant, you will see that it is the 
normalized polynomial that is fed to the resultant computation:

(%i6) :lisp(trace resultant)

(RESULTANT)
(%i6) poly_discriminant(x^2/2-x+3, x);
   0: (MAXIMA::RESULTANT (#:X427 2 1 1 -2 0 6) (#:X427 1 2 0 -2))
   0: RESULTANT returned 20
(%o6)                                - 20

In this trace, (#:X427 2 1 1 -2 0 6) means x^2 -2*x +6 and (#:X427 1 2 0 
-2) means 2*x-2.  Generally a polynomial is represented

as a list (n,a_n, ....,1,a_1,0,a_0) with the variable in front.

At what point has the polynomial been normalized i don't know. Anyways, 
i don't think it is a good idea to do any form of quotienting,

in particular because we may be dealing with several variables 
polynomials, then dividing by the leading coefficient may produce a big 
mess.

It is precisely to avoid such divisions that one uses pseudo division 
instead of straight euclidean division in many computations, particularly

in the subresultant business.  In fact the real originality in the Brown 
and Traub algorithm is to find an "optimal" way of dealing with these

leading coefficients problems. To get a grasp of the computational 
problem, you can look at

https://en.wikipedia.org/wiki/Polynomial_greatest_common_divisor

notably at the end, the section pseudo remainder sequences, which is 
very well done.  Anyways in my opinion resultants and so on

are defined up to a constant, and it makes no sense to say that the 
correct answer for the discriminant is this or that. For example

the resultant(f,g) is defined as the condition for f and g to have a 
common root and this leads to the determinant formula you mention.

But this is obviously up to a constant.


-- 
Michel Talon