Solving polynomial systems with homotopy. Where is the bottleneck?
$begingroup$
I have a system of polynomial equations with $n$ unknowns (where $n$ can be between 3 and 20) and that is known to have at least $n!$ isolated solutions.
I want to solve this system numerically, but if I plug it in an algebraic computing system (Macaulay2 or Bertini) and try to solve it with homotopy continuation, the PC just remains blocked, and I can't find any solutions.
I am trying to understand why this happens, with the following consideration: all the $n!$ solutions are in some way equivalent (I mean that if $(x_1,..x_n)$, than the other solutions are just permutations of this array), so I am interested in computing just $1$ solution, not all the $n!$ solutions. My questions:
- Where is the bottleneck of the homotopy method, in the number of solutions or in the number of variables?
- The fact that I am interested only in one solution, may speed up the method just analyzing a single homotopy path?
- Is there a way in Macaulay2 to calculated only a single solution, and not all of them?
Thanks!
algebraic-geometry polynomials systems-of-equations macaulay2
edited Dec 8 '18 at 23:49
Rodrigo de Azevedo
12.9k41856
asked Mar 22 '16 at 9:57
Ulderique DemoitreUlderique Demoitre
19810
$endgroup$
add a comment |
$begingroup$
I have a system of polynomial equations with $n$ unknowns (where $n$ can be between 3 and 20) and that is known to have at least $n!$ isolated solutions.
I want to solve this system numerically, but if I plug it in an algebraic computing system (Macaulay2 or Bertini) and try to solve it with homotopy continuation, the PC just remains blocked, and I can't find any solutions.
I am trying to understand why this happens, with the following consideration: all the $n!$ solutions are in some way equivalent (I mean that if $(x_1,..x_n)$, than the other solutions are just permutations of this array), so I am interested in computing just $1$ solution, not all the $n!$ solutions. My questions:
- Where is the bottleneck of the homotopy method, in the number of solutions or in the number of variables?
- The fact that I am interested only in one solution, may speed up the method just analyzing a single homotopy path?
- Is there a way in Macaulay2 to calculated only a single solution, and not all of them?
Thanks!
algebraic-geometry polynomials systems-of-equations macaulay2
edited Dec 8 '18 at 23:49
Rodrigo de Azevedo
12.9k41856
asked Mar 22 '16 at 9:57
Ulderique DemoitreUlderique Demoitre
19810
$endgroup$
$begingroup$
Over what field?
$endgroup$
– Rodrigo de Azevedo
Dec 8 '18 at 23:49
add a comment |
$begingroup$
I have a system of polynomial equations with $n$ unknowns (where $n$ can be between 3 and 20) and that is known to have at least $n!$ isolated solutions.
I want to solve this system numerically, but if I plug it in an algebraic computing system (Macaulay2 or Bertini) and try to solve it with homotopy continuation, the PC just remains blocked, and I can't find any solutions.
I am trying to understand why this happens, with the following consideration: all the $n!$ solutions are in some way equivalent (I mean that if $(x_1,..x_n)$, than the other solutions are just permutations of this array), so I am interested in computing just $1$ solution, not all the $n!$ solutions. My questions:
- Where is the bottleneck of the homotopy method, in the number of solutions or in the number of variables?
- The fact that I am interested only in one solution, may speed up the method just analyzing a single homotopy path?
- Is there a way in Macaulay2 to calculated only a single solution, and not all of them?
Thanks!
algebraic-geometry polynomials systems-of-equations macaulay2
edited Dec 8 '18 at 23:49
Rodrigo de Azevedo
12.9k41856
asked Mar 22 '16 at 9:57
Ulderique DemoitreUlderique Demoitre
19810
$endgroup$
I have a system of polynomial equations with $n$ unknowns (where $n$ can be between 3 and 20) and that is known to have at least $n!$ isolated solutions.
I want to solve this system numerically, but if I plug it in an algebraic computing system (Macaulay2 or Bertini) and try to solve it with homotopy continuation, the PC just remains blocked, and I can't find any solutions.
I am trying to understand why this happens, with the following consideration: all the $n!$ solutions are in some way equivalent (I mean that if $(x_1,..x_n)$, than the other solutions are just permutations of this array), so I am interested in computing just $1$ solution, not all the $n!$ solutions. My questions:
- Where is the bottleneck of the homotopy method, in the number of solutions or in the number of variables?
- The fact that I am interested only in one solution, may speed up the method just analyzing a single homotopy path?
- Is there a way in Macaulay2 to calculated only a single solution, and not all of them?
Thanks!
algebraic-geometry polynomials systems-of-equations macaulay2
algebraic-geometry polynomials systems-of-equations macaulay2
edited Dec 8 '18 at 23:49
Rodrigo de Azevedo
12.9k41856
asked Mar 22 '16 at 9:57
Ulderique DemoitreUlderique Demoitre
19810
edited Dec 8 '18 at 23:49
Rodrigo de Azevedo
12.9k41856
asked Mar 22 '16 at 9:57
Ulderique DemoitreUlderique Demoitre
19810
edited Dec 8 '18 at 23:49
Rodrigo de Azevedo
12.9k41856
edited Dec 8 '18 at 23:49
Rodrigo de Azevedo
12.9k41856
edited Dec 8 '18 at 23:49
Rodrigo de Azevedo
12.9k41856
12.9k41856
asked Mar 22 '16 at 9:57
Ulderique DemoitreUlderique Demoitre
19810
asked Mar 22 '16 at 9:57
Ulderique DemoitreUlderique Demoitre
19810
asked Mar 22 '16 at 9:57
Ulderique DemoitreUlderique Demoitre
19810
19810
$begingroup$
Over what field?
$endgroup$
– Rodrigo de Azevedo
Dec 8 '18 at 23:49
add a comment |
$begingroup$
Over what field?
$endgroup$
– Rodrigo de Azevedo
Dec 8 '18 at 23:49
$begingroup$
Over what field?
$endgroup$
– Rodrigo de Azevedo
Dec 8 '18 at 23:49
$begingroup$
Over what field?
$endgroup$
– Rodrigo de Azevedo
Dec 8 '18 at 23:49
add a comment |
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$begingroup$
Over what field?
$endgroup$
– Rodrigo de Azevedo
Dec 8 '18 at 23:49