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Eine Frage!
https://www.singular.uni-kl.de/forum/viewtopic.php?f=10&t=1856		 Page 1 of 1
Author:  LDT [ Wed Sep 08, 2010 8:36 pm ]
Post subject:  Eine Frage!
Kann Singular die Frage nach der Existenz von Lösungen eines Gleichungssystems nach gewisser Körpererweiterung beanworten?

z.b habe ich 2 Gleichungen:

(1) x^2 - tu^2 + t = ( t^2u^2 - t )y^2 \neq 0

(2) x^2 - 2tu^2 + 1/t = t(t^2u^2 - t)z^2 \neq 0

über Q(t) oder C(t) definiert, wobei x,y,z,u Unbestimmte sind. Ich wollte mal wissen ob das system nach einer Körpererweiterung vom ungeraden Grad, spricht 3,5,7...eine Lösung hat. Kann ich das dann mit Singular programmieren?
Author:  LDT [ Fri Sep 10, 2010 7:25 am ]
Post subject:  Re: Eine Frage!
No one want to reply me? :( i should post then my question once again.

Suppose we are given a system of equations, say

(1) x^2 - tu^2 + t = ( t^2u^2 - t )y^2 \neq 0

(2) x^2 - 2tu^2 + 1/t = t(t^2u^2 - t)z^2 \neq 0

where x,y,z,u are variables and these equations are defined over C(t). Can Singular tell me if i have a solution after a field extension of degree odd?
Author:  gorzel [ Wed Sep 15, 2010 2:31 pm ]
Post subject:  Re: Eine Frage!
Your question is somehow selfcontrdaticting:

With "\neq 0" it is not an equation.

Supposed that you mean "=0", then see

System of polynomial equations
viewtopic.php?f=10&t=1801&start=0

where a problem similar to yours is discussed.
Author:  LDT [ Wed Sep 15, 2010 8:38 pm ]
Post subject:  Re: Eine Frage!
No there is no contradiction here. The last \neq 0 means that you consider an open dense Zariski subset of the affine variety defined by the equations.
Author:  gorzel [ Fri Sep 17, 2010 2:51 pm ]
Post subject:  Re: Eine Frage!
LDT wrote:
No there is no contradiction here. The last \neq 0 means that you consider an open dense Zariski subset of the affine variety defined by the equations.


To clarify your problem, let us recall first the mathematical terminology:

...=0 is called an equation

..\neq 0 is called an inequality, (Ungleichung in German).
Author:  Guest [ Fri Sep 17, 2010 10:40 pm ]
Post subject:  Re: Eine Frage!
When i wrote

U = {f(x,y,z,u) = g(x,y,z,u) \neq 0 }

it means that U is an open subset of

X = {f(x,y,z,u) = g(x,y,z,u)}.
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