Singular

Here is a counter example:

f=x^2*(x-1)^2*(y^2+1)+y2=0 defines two points (0,0),(1,0) in R^2 which are the union of x=y=0 and x=1,y=0,
but f is irreducible in R[x,y], even in C[x,y] that is V(f) is irreducible in C^2 in the Zariski topology.
Adding squares of new variables makes this an example in any R^n, n>=2.

See the discussion under the topic "A problem in algebraic geometry" in this forum:
http://www.singular.uni-kl.de/forum/viewtopic.php?f=10&t=1816

Gert-Martin

Sorry I did not comment to this sooner I hope this clarifyes my question.

I thought this question were complitely forgotten. Many thank's for answering to this one. I am not exactly an expert in alg.geom but it is one of the ''tools''
that I use.

And let me say here: You guys have done wonderful job with Singular. I own you more than you can imagine. Otherwise I would have to had settle with Macaulay 2 or CoCoa :-))).

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I am just asking that if I=<f_1,...f_n>\in C[x1,...,xn] and V(I) is irreducible in Zariski topology

then if you change the the ring variables (x1,...xn) to reals (assuming that the multiplyers of monomials of generators f_i are real)

then I=<f_1,...,f_n>\in R[x1,...xn] and V(I) is also irreducible in Zariski topology.

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So basically I am asking if a variety V(I)\in C^n can not be presented as a nontrivial union of two varieties is it true that

then V(I) \in R^n can not be presented as a nontrivial union of two subvarieties (when you change the ring variables to reals).

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Probably this is a very trivial question, but I have lot's of other things on my mind right now. If the question is still unclear just let me know I try to answer you asap.

I do not understand your conjecture: Do you have K=R or K=C? Do you want to consider V(I) intersected with R, since V(I) is - at least in the beginning of the statement - a subset of C, or is it not? Please clarify.
Frank

I got lot's of help from here last time. I know there are lot's of people here who knows about alg.geom & commut. alg.

I have only have one question which has bothered me and don't just have
time to think this one through

Is the following conjecture true or false

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Conjecture 1.

Let I=<p_1,..,p_n>\subset\K[x_1,...,x_n] be an ideal

If K=C and V(I)\subset\C^n is irreducible in Zariski topology then ---> If K=R V(I)\subset\R^n is also irreducible in Zariski topology

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I just did not have time for this and don't have any books available.
If somebody knows this result or a counterexample it would be grately appreciated !

Zariski topology is defined by declearing open sets to complements of affine varieties.