Doing proofs with the variety function and the Zariski topology

In summary: What you are asking for is not provided. What you wrote was rigorous, the notations and the phrasing are unimportant. If you insist just write it with the notations you prefer. Let ##p\in V(\cup A_i)##, then for all ##i## and all ##f\in A_i##, we have that ##f(p)=0##. Thus ##p\in V(A_i)## for all ##i##.
  • #1
Mikaelochi
40
1
TL;DR Summary
Problem is shown in the image
20211020_134055.jpg

I included this image because it is easier than typing it out. Anyway, this is an old problem I need to catch up on. I have a clue as to how to do part a. I could say given an x that is a member of ∩V(Ai) which implies that x is a member of V(Ai) for ∀i. Then we can say ∀i all polynomials are in Ai when the polynomial is equal to zero at x. Apparently this statement is the same as V of the union of Ai. Still a little hazy on that. I don't know how to show the converse is true (which would prove the equivalency). This problem has me quite lost. But I suspect (b), (c), and (d) follow nicely from understanding (a). Any help is greatly appreciated. Thanks!
 
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  • #2
##V(A)## means the common zeroes of all polynomials in ##A##. The right hand side is the common zeroes of all polynomials in all ##A_i##. The left hand side says the same.
 
  • #3
martinbn said:
##V(A)## means the common zeroes of all polynomials in ##A##. The right hand side is the common zeroes of all polynomials in all ##A_i##. The left hand side says the same.
I know that, I'm just having some trouble showing the equivalency of problem (a)
 
  • #4
Mikaelochi said:
I know that, I'm just having some trouble showing the equivalency of problem (a)
But that is the showing of the equivalency!
 
  • #5
martinbn said:
But that is the showing of the equivalency!
I mean I have to use logic notation and probably some ideas from set theory to rigorously prove this is the case
 
  • #6
Mikaelochi said:
I mean I have to use logic notation and probably some ideas from set theory to rigorously prove this is the case
What I wrote was rigorous, the notations and the phrasing are unimportant. If you insist just write it with the notations you prefer. Let ##p\in V(\cup A_i)##, then for all ##i## and all ##f\in A_i##, we have that ##f(p)=0##. Thus ##p\in V(A_i)## for all ##i##.
 

1. What is the variety function in mathematics?

The variety function, also known as the algebraic variety function, is a mathematical tool used to describe the solution set of a system of polynomial equations over a given field. It maps the set of polynomial equations to the set of points that satisfy those equations.

2. What is the significance of the Zariski topology in doing proofs with the variety function?

The Zariski topology is a topology defined on the set of solutions of a system of polynomial equations. It is used in conjunction with the variety function to study the geometric and topological properties of algebraic varieties. It allows for the classification and comparison of different varieties based on their topological properties.

3. How is the variety function used in algebraic geometry?

The variety function is a fundamental tool in algebraic geometry, as it allows for the study and classification of algebraic varieties. It is used to define and characterize geometric objects such as curves, surfaces, and higher-dimensional varieties in terms of their underlying polynomial equations.

4. Can the variety function and the Zariski topology be applied to any system of polynomial equations?

Yes, the variety function and the Zariski topology can be applied to any system of polynomial equations over any field. However, they are most commonly used in the context of algebraic geometry, where the field is typically the complex numbers.

5. What are some common applications of the variety function and the Zariski topology?

The variety function and the Zariski topology have various applications in mathematics and other fields such as physics and computer science. They are used to study algebraic varieties, solve systems of polynomial equations, and classify geometric objects. They are also used in cryptography and coding theory to construct error-correcting codes.

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