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| How to compute critical values? https://www.singular.uni-kl.de/forum/viewtopic.php?f=10&t=2015 |
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| Author: | default1 [ Tue Nov 22, 2011 10:08 am ] |
| Post subject: | How to compute critical values? |
Hi, i want to compute the critical values of different polynoms in three variables (x,y,z). Usually the following steps work really fine: LIB "primdec.lib"; ring r=0,(x,y,z,t),dp; poly f = 3*x^2-3*x*y^2+y^3-3*y+3*z^2; ideal i = jacob(f),f-t; primdecGTZ(i); Normally i get something, that starts with an expression like: t=a, so i can see the critical value is a. In the example case from above, i get: t^4+24*t^2-48. how can i determine the critical values in this case? Do i have to change the ordering? Or does somebody have a better idea, how to compute the critical values? Please help! Greetings, default1 |
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| Author: | gorzel [ Wed Nov 23, 2011 12:42 am ] |
| Post subject: | Re: How to compute critical values? |
You want to compute the critical values of a polynomial mapping from C^3-> C. The primary decomposition is not what you really need to compute. You have just to eliminate the variables x,y,z from the ideal. Code: > ring r=0,(x,y,z,t),dp; > poly f = 3*x^2-3*x*y^2+y^3-3*y+3*z^2; > ideal i = jacob(f),f-t; > eliminate(i,xyz); _[1]=64t3+176t2-24t-665 This polynomial is irreducible over Q. There are three isolated singularities each with Milnor number 1 Code: > ring rdp =0,(x,y,z),dp; > poly f = 3*x^2-3*x*y^2+y^3-3*y+3*z^2; > vdim(std(jacob(f))); 3 The coordinates (x,y,z) expressed in terms by t can be read off from the primdec result. Code: > setring r;
> primdecSY(i); [1]: [1]: _[1]=64t3+176t2-24t-665 _[2]=z _[3]=575y-16t2+248t+80 _[4]=1150x-144t2-68t+145 |
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