Singular

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?

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?

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.

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).

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)}.