Except if has degree one, this problem is very much open. But it makes sense to translate it to a more geometric setting of polynomials over finite fields, and this leads (as is often the case) to problems that are more tractable. The translation is straightforward: instead of , one considers the ring of polynomials over a finite field with elements, instead of , one considers a polynomial , and then the question is to determine asymptotically how many polynomials of given degree are such that is an irreducible polynomial.
The reason the problem becomes more accessible is that there is an algebraic criterion for a polynomial with coefficients in a finite field to be irreducible: if we look at the natural action of the Frobenius automorphism on the set of roots of the polynomial, then is irreducible if and only if this action “is” a cycle of length . This is especially useful for the variant of the Schinzel problem where the size of the finite field is varying, whereas the degree of the polynomials remains fixed, since in that case the variation of the action of the Frobenius on the roots of the polynomial is encoded in a group homomorphism from the Galois group of the function field of the parameter space to the symmetric group on letters. (This principle goes back at least to work of S.D. Cohen on Hilbert’s Irreducibility Theorem).
If we apply this principle in the Schinzel setting, this means that we consider specialized polynomials for some fixed polynomial , where runs over polynomials of a fixed degree , but ranges over powers of a fixed prime. “Generically”, the polynomial has some fixed degree , and is squarefree. If we interpret the parameter space geometrically, the content of the previous paragraph is that we have a group homomorphism
from the fundamental group of to the symmetric group. Then the Chebotarev Density Theorem solves, in principle, the problem of counting the number of irreducible specializations in the large limit: essentially (omitting the distinction between geometric and arithmetic fundamental groups), the asymptotic proportion of such that is irreducible converges as to the proportion, in the image of , of the elements that are -cycles in . If the homomorphism is surjective, then this means that the probability that is irreducible is about . This is the expected answer in many cases, because this is also the probability that a random polynomial of degree is irreducible.
All this has been used by a number of people (including Hall, Pollack, Bary-Soroker, and most successfully Entin). However, there is a nice geometric interpretation that I haven’t seen elsewhere. To see it, we go back to and the action of Frobenius on its roots that will determine if is irreducible. A root of is an element such that
where we view as a two-variable polynomial. In other words, is the first coordinate of a point that belongs to the intersection of the graph of in the plane, and the plane affine curve with equation . Since the Frobenius will permute these intersection points in the same way that it permutes the roots of , we can interpret the Schinzel Problem, in that context, as asking about the “variation” of this Galois action as varies and the curve is fixed.
This point of view immediately suggests some generalizations: there is no reason to work over a finite field (any field will do), the base curve (which is implicitly the affine line where polynomials live) can be changed to another (open) curve ; the point at infinity, where polynomials have their single pole, might also be changed to any effective divisor with support the complement of in its smooth projective model (e.g., allowing poles at and on the projective line); and may be any (non-vertical) curve in . For instance (to see that this generalization is not pointless), take any curve , and define . Then the intersection of the graph of a function on and is the set of zeros of . The problem becomes something like figuring out the “generic” Galois group of the splitting field of this set of zeros. (E.g., the Galois group of a complicated elliptic function defined over …)
In fact this special case was (with different motivation and terminology) considered by Katz in his book “Twisted L-functions and monodromy” (see Chapter 9). Katz shows that if the (fixed) effective divisor used to define the poles of the functions considered has degree , where is the genus of the smooth projective model of , then the image of Galois is the full symmetric group (his proof is rather nice, using character sums on the Jacobian…)
The general case, on the other hand, does not seem to have been considered before. In the recent note that I’ve written on the subject, I use quite elementary arguments with Lefschetz pencils / Morse-like functions (again inspired by results of Katz and Katz-Rains) to show that in very general conditions, the image of the fundamental group is again the full symmetric group. This gives the asymptotic for this geometric Schinzel problem in this generality over finite fields. (In the classical case, this was essentially done by Entin, though the conditions of applicability are not exactly the same).
I recently gave a talk about this in Berlin, and the slides might be a good introduction to the ideas of the proof for interested readers…
As I mention at the end of those slides, the next step is of course to think about the fixed finite field case, where the degree of the polynomials tends to infinity. This seems, even geometrically, to be quite an interesting problem…
[Update: after I wrote this post, I remembered that in fact the (qualitative) problem of representing primes with one polynomial that I consider here is actually Bunyakowski’s Problem, and that the Schinzel Hypothesis is the qualitative statement for a finite set of polynomials… The quantitative versions of both are usually called the Bateman-Horn conjecture. So my terminology is multiply inaccurate…]
]]>Consider an open disc contained in the region (other compact regions may be considered, for instance an open rectangle). For any real number , we can look at the function on . This is a holomorphic function on , continuous on the closed disc . What kind of functions arise this way? Bagchi proved the following (this is essentially Theorem 3.4.11 in his thesis):
Theorem. Let denote the Banach space of holomorphic functions on which are continuous on the closed disc. For , define a probability measure on to be the law of the random variable , where is uniformly distributed on . Then converges in law, as , to the random holomorphic function
,
where is a sequence of independent random variables indexed by primes, all uniformly distributed on the unit circle.
This is relatively easy to motivate: if we could use the Euler product
in , then we would be led to an attempt to understand the probabilistic behavior of the sequence , viewed as a random variable on with values in the infinite product of copies of the unit circle indexed by primes. This is a compact topological group, and the easy answer (using the Weyl criterion) is simply that this sequence converges to the Haar measure on . In other words, the random sequence converges in law to a sequence of independent, uniform, random variables on the unit circle. Then it is natural to expect that should converge to the random function , which is obtained formally by replacing by its limit .
Bagchi’s proof is somewhat intricate, in comparison with this heuristic justification, especially if one notices that if is replaced by a compact region in the domain of absolute convergence, then the same idea applies, and is a completely rigorous proof (one need only observe that the assignment of an Euler product
to a sequence of complex numbers of modulus one is a continuous operation in the region of absolute convergence.)
The proof I give in my script tries to remain closer to the basic intuition, and is indeed less involved (it avoids both a use of the pointwise ergodic theorem that Bagchi required and any use of tightness or weak-compactness). It makes it easy to see exactly what arithmetic ingredients are needed, beyond the convergence in law of to the Haar measure on . Roughly speaking, it goes as follows:
Now pick some parameter , and write
,
where
Fix . For some fixed big enough, is less than by Step 3, and is at most . For this fixed , tends to as tends to infinity because of the convergence in law of to — the sum defining the truncations are finite, so there is no convergence issue. So for all large enough, we will get
proudly as French and German books, or going down
as English or American or Italian books? When books are ordered by topic or author, this leads to rather uncomfortable switches of orientation of the head as one scans bookshelves for the right oeuvre to read during a lazy afternoon.
Actually, these are more or less contemporary examples, and it seems that these conventions change with time. For instance, I have an old English paperback from 1951 where the title goes up instead of down:
Another from 1962 goes down. When did the change happen? And why? And how do other languages stack up? Is it rather a country-based preference? Are the titles of Italian-language books printed in Switzerland going up (like the French and German ones do), or down? And does this affect the direction in which shivers run along your spine when reading a scary story of murdered baronets in abandoned ruins?
(There’s of course the solution, admittedly snobbish, of writing the title and author’s name horizontally
as the Pléiade does, for instance).
]]>The email address from which it came ( jlb@limbo.ow ) is probably not genuine, so I wonder who the author could be (the final note “Translated, from the Spanish, by H.A.H” is of course suggestive, but one would then like to see the original Spanish…)
]]>The meeting is organized by the Forschungsinstitut für Mathematik, and (again as usual!) there is a certain amount of funding for local expenses made available by FIM for young researchers (graduate students and postdocs). Please register on the FIM web page before March 21 if you are interested!
]]>The negative comments embody the perfect distillation of the perverse puritanical hair-splitting competition known as “Is this question a good fit for MO?” (to be read in a slightly hysterical voice) that is now what I find most annoying on the site. This is not what mathematics (not even “research” mathematics, that seems to replace here the “pure” mathematics illusion of yesteryears) is about for me. I must confess to finding particularly annoying that some of the most vocal critics (e.g., the pseudonymous “quid”) seem to be people with little actual mathematical contributions and too much time to spend and to write for ever and ever on the finer points of etiquette of a web site as if it were some platonic object to protect from all interlopers.
What would Arnold think of this discussion, where “mathematicians” throw away much (he would say “most”) of the whole history, motivation and insights of their science? Would a question of Kolmogorov on what the brain looks like as graph have passed through the fourches caudines of Signor Quid?
]]>Roughly speaking, I concentrated on a small number of examples of statements of convergence in law of sequences of arithmetically-defined probability measures: (1) the Erdös-Kac Theorem; (2) the Bohr-Jessen-Bagchi distribution theorems for the Riemann zeta function on the right of the critical line ; (3) Selberg’s Theorem on the normal behavior of the Riemann zeta function on the critical line ; (4) my own work with W. Sawin on Kloosterman paths. My goal in each case was to do with as little arithmetic, and as much probability, as possible. The reason for this is that I wanted to get and give as clear a picture as possible of which arithmetic facts are involved in each case.
This led to some strange and maybe pedantic-looking discussions at the beginning. But overall I think this was a useful point of view. I think it went especially well in the discussion of Bagchi’s Theorem, which concerns the “functional” limit theorem for vertical translates of the Riemann zeta function in the critical strip, but to the right of the critical line. The proof I gave is, I think, simpler and better motivated than those I have seen (all sources I know follow Bagchi very closely). In the section on Selberg’s Theorem, I used the recent proof found by Radziwill and Soundararajan, which is very elegant (although I didn’t have much time to cover full details during the class).
I have started writing notes for the course, which can be found here. Note that, as usual, the notes are potentially full of mistakes! But those parts which are written are fairly complete, both from the arithmetic and probabilistic point of view.
]]>The basic strategy to get this result is not new: it was devised by Fouvry and Michel a number of years ago (inspired at least in part by earlier work of Friedlander-Iwaniec and by the Vinogradov-Karatsuba-style “shift” method to estimate certain short exponential sums). What was missing (despite the strong motivation provided by applications that were known to follow from such a result, one of which is described in a recent preprint of Blomer, Fouvry, Milicevic, Michel and myself) was a way to prove certain estimates for (complete) sums over finite fields, of the type
where
unless the parameters are in some “diagonal” positions. And we cannot afford too many diagonal cases…
The main contribution of our paper (much of which comes from the ideas of Will!) is to find a relatively robust approach to such estimates.
This relies, as one can expect, from extensive algebraic-geometric arguments to apply the Riemann Hypothesis over finite fields. In fact, from this point of view, this paper is by far the most complicated I’ve ever been involved in. We use, among other things:
Many of these are results and ideas that I was aware of but had never actually used before, and I learnt a lot by seeing how Will exploited and combined them. I will try to write a few more posts later to (attempt to) explain and motivate them (and how we use them) from an analytic nunber theorist’s viewpoint. The theory of vanishing cycles, in particular, should have many more applications in extending the range of applicability of Deligne’s Riemann Hypothesis to problems in analytic number theory.
The paper is dedicated to Henryk Iwaniec, who has been over the years the most eloquent and powerful advocate for a deeper use of the work of Deligne (and Katz and others) in applications to analytic number theory.
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