[Maxima-bugs] [ maxima-Bugs-1651948 ] algsys/solve cannot find solutions

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Bugs item #1651948, was opened at 2007-02-04 14:07
Message generated for change (Settings changed) made by robert_dodier
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>Category: Lisp Core - Solving equations
Group: None
Status: Open
Resolution: None
Priority: 5
Private: No
Submitted By: Nobody/Anonymous (nobody)
Assigned to: Nobody/Anonymous (nobody)
Summary: algsys/solve cannot find solutions

Initial Comment:
(%i169) p1:-x*y^3+y^2+x^4-9*x/8$
(%i170) p2:y^4-x^3*y-9*y/8+x^2$

(%i180) algsys([p1,p2],[x,y]);
(%o180) [[x = 1/2,y = 1],[x = 9/8,y = 9/8],[x = 1,y = 1/2],[x = 0,y = 0]]

the system seems to have 4 (real) solutions.

`? algsys' says:
	...
     The method is as follows:

     (1) First the equations are factored and split into subsystems.

     (2) For each subsystem <S_i>, an equation <E> and a variable <x>
     are selected.  The variable is chosen to have lowest nonzero
     degree.  Then the resultant of <E> and <E_j> with respect to <x>
     is computed for each of the remaining equations <E_j> in the
     subsystem <S_i>.  This yields a new subsystem <S_i'> in one fewer
     variables, as <x> has been eliminated.  The process now returns to
     (1).
	...

so `algsys' uses the resultant of the two polynomial with respect to x or y. i do this by hand:

(%i184) factor(resultant(p1,p2,x));
(%o184) 4096*(y-1)*y*(2*y-1)*(8*y-9)*(y^2+y+1)*(4*y^2+2*y+1)*(64*y^2+72*y+81)

(resultant with respect to x gives a similar result)
this gives  6 additional solution for y that can be found by sove. if i substitute such an y in the original polyomial the resulting polynomials in x have degree 4 and are also solvable. why don't `algsys' or `solve' don't find these solution?


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Comment By: Barton Willis (willisbl)
Date: 2007-02-05 05:45

Message:
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user_id=895922
Originator: NO

Here is a workaround for your equations; the workaround might help in
general:

(%i34) load(grobner)$
Loading maxima-grobner $Revision: 1.2 $ $Date: 2006/11/08 03:40:02 $
(%i35) p1:-x*y^3+y^2+x^4-9*x/8$
(%i36) p2:y^4-x^3*y-9*y/8+x^2$
(%i37) eqs : map('ratnumer, [p1,p2])$
(%i38) eqs : poly_reduced_grobner(eqs,[x,y])$
(%i39) algsys(eqs,[x,y]);
(%o39)
[[x=0,y=0],[x=1,y=1/2],[x=9/8,y=9/8],[x=1/2,y=1],[x=(sqrt(3)*%i-1)/4,y=-(sqrt(3)*%i+1)/2],[x=-
(sqrt(3)*%i+1)/4,y=(sqrt(3)*%i-1)/2],[x=(9*sqrt(3)*%i-9)/16,y=-(9*sqrt(3)*%i+9)/16],[x=-(9*sqrt(3)*%i+9)/16,y=(9*sqrt(3)*%i-9)/16],[x=(sqrt(3)*%i-1)/2,y=-
(sqrt(3)*%i+1)/4],[x=-(sqrt(3)*%i+1)/2,y=(sqrt(3)*%i-1)/4]]



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