Irreducible algebraic sets via irreducible polynomials

Question

There are many results about irreducible polynomials over finite fields:

we know a cardinality of all irreducible polynomials with given degree, we know explicit examples of irreducible polynomials, we know effective algorithms for construction irreducible polynomials, and for verifying that given polynomial is irreducible.

Are there similar results about irreducible algebraic sets? Could you give a reference?

Thank you!

UPD: by explicit examples of irreducible polynomials I mean the following: $x^p - x + a$ is irreducible over $\mathbb{F}_q$ if $a \not= 0$. Are there are similar examples for an arbitrary dimension?

I want to have such explicit examples of irreducible algebraic sets because I hope it can help to solve my another question: an algebraic variety for a boolean circuit

Answer 1

Bjorn Poonen and his coauthors have studied topics of this type in a series of papers on "Bertini Theorem over Finite Fields".

See for example the paper:

Poonen, Charles - Bertini irreducibility theorems over finite fields.

http://www-math.mit.edu/~poonen/papers/bertini_irred.pdf

Edit: I should clarify; this answer addresses the first part of the question, which asks about generalisations of "cardinality of all irreducible polynomials with given degree".

Answer 2

Are there are similar examples for an arbitrary dimension?

I am not sure exactly what this question means, but it sounds like you are asking for examples of families of irreducible algebraic sets.

If that is your question, then $\mathrm{SL}_n$ is an irreducible affine algebraic group since the determinant is an irreducible polynomial.

More generally, any connected affine algebraic group $G$ is irreducible. You can find many examples of "arbitrary dimension" there.

Although looking at examples is generally a good practice, irreducible algebraic sets are fairly common place, and so might be too large of a class to examine for your particular problem.

Maybe a better question would be:

What general methods exist to determine if an algebraic set is irreducible?

For example, over $\mathbb{C}$ an affine algebraic set is irreducible if and only if it contains a dense open path-connected subset of smooth points (in the strong topology).