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parametrization with respect to given variables
https://www.singular.uni-kl.de/forum/viewtopic.php?f=10&t=1316		 Page 1 of 1
Author:  SingularTeam [ Thu Aug 11, 2005 5:31 pm ]
Post subject:  parametrization with respect to given variables
I have 3 polynomials with the variables {s13, h13, h123,
h23, s12, h12, c, h1, h2, h3, s1}. Please see if you can express from
this the variables {s1,s12, s13} in terms of the others.

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Posted in old Singular Forum on: 2002-02-03 10:08:32+01
Author:  G.-M. Greuel [ Thu Aug 11, 2005 5:42 pm ]
Post subject:  Re: parametrization with respect to given variables
For this example primdecGTZ is too expensive, however, minAssGTZ works
(here also triangMH or triangM seem to work too). For your purpose one has
to declare h13,h123,h23,h12,c,h1,h2,h3 as parameters and s1,s12,s13 as
variables:

option(prot);
ring r = (0,h13,h123,h23,h12,c,h1,h2,h3),(s1,s12,s13),lp;
ideal i =
12*h23*s1^2*s13^2+12*h13*s1^3*s13+2*h23*s12*s1^3-10*h23*s1^2*s13
...
+h3*s12*s13+h123*s12*s1+h123*s12*s13;

list ma = minAssGTZ(i); //the minimal associated primes
ma;
[1]:
_[1]=s13
_[2]=s1
[2]:
_[1]=(-2*c^2)*s13+(h13^2+h13*h123+2*h13*h23+h13*h1+h13*h2+h13*h3+h123*h23+h123*h1+h123*h2+h23^2+h23*h1+h23*h2+h23*h3+h1*h3+h2*h3)
_[2]=(-h13*c^2-h123*c^2-h23*c^2+c^3-c^2*h1-c^2*h2-c^2*h3)*s12+(-h13^2*h12-h13*h123*h12-2*h13*h23*h12+2*h13*h12*c-h13*h12*h1-h13*h12*h2-h13*h12*h3-h123*h23*h12
+h123*h12*c-h123*h12*h1-h123*h12*h2-h23^2*h12+2*h23*h12*c-h23*h12*h1-h23*h12*h2-h23*h12*h3-h12*c^2+h12*c*h1+h12*c*h2+h12*c*h3-h12*h1*h3-h12*h2*h3)
_[3]=(-2*c^2)*s1+(-h13^2-h13*h123-2*h13*h23+h13*c-h13*h1-h13*h2-h13*h3-h123*h23-h123*h1-h123*h2-h23^2+h23*c-h23*h1-h23*h2-h23*h3+c*h1+c*h2-h1*h3-h2*h3)
[3]:
_[1]=s12
_[2]=2*s1+2*s13-1
[4]:
_[1]=s12
_[2]=s1

//Or, if you wish to normalize the second solution, then you get the expression for
//s13,s12 and s1 directly (of course, you get the solutions by setting the
//polynomials 0):

normalize(ma[2]);
_[1]=s13+(-h13^2-h13*h123-2*h13*h23-h13*h1-h13*h2-h13*h3-h123*h23-h123*h1-h123*h2-h23^2-h23*h1-h23*h2-h23*h3-h1*h3-h2*h3)/(2*c^2)
_[2]=s12+(h13^2*h12+h13*h123*h12+2*h13*h23*h12-2*h13*h12*c+h13*h12*h1+h13*h12*h2+h13*h12*h3+h123*h23*h12-h123*h12*c+h123*h12*h1+h123*h12*h2+h23^2*h12-2*h23*h12*c
+h23*h12*h1+h23*h12*h2+h23*h12*h3+h12*c^2-h12*c*h1-h12*c*h2-h12*c*h3+h12*h1*h3+h12*h2*h3)/(h13*c^2+h123*c^2+h23*c^2-c^3+c^2*h1+c^2*h2+c^2*h3)
_[3]=s1+(h13^2+h13*h123+2*h13*h23-h13*c+h13*h1+h13*h2+h13*h3+h123*h23+h123*h1+h123*h2+h23^2-h23*c+h23*h1+h23*h2+h23*h3-c*h1-c*h2+h1*h3+h2*h3)/(2*c^2)

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Posted in old Singular Forum on: 2002-02-03 10:10:53+01
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