Joint distribution of conjugate algebraic numbers: a random polynomial approach
Given a polynomial and a vector of positive weights , define the -weighted -norm of as
In the non-weighted case , this notion generalizes the naïve height (), the length (), and the Euclidean norm (). An important weighted example is the Bombieri -norm:
Define the -weighted -norm of an algebraic number to be the -weighted -norm of its minimal polynomial. For non-negative integers such that and a Borel subset denote by the number of ordered -tuples in of conjugate algebraic numbers of degree and -weighted -norm at most . We show that
where is the volume of the unit -weighted -ball and shall denote the correlation function of real and complex zeros of the random polynomial for i.i.d. random variables with density for resp. with constant density on for . We give an explicit formula for which in the case simplifies to
where is the monic polynomial whose zeros are the arguments of the correlation function and denotes its discriminant.
As an application, we show that with respect to the Bombieri 2-norm, i.e. for and , the asymptotic density of real algebraic numbers coincides with the normalized Cauchy density up to a constant factor depending on only:
Key words and phrases:conjugate algebraic numbers, correlations between algebraic numbers, distribution of algebraic numbers, integral polynomials, random polynomials, mixed correlation functions, real zeros, complex zeros, Coarea formula, Bombieri norm, -balls, weighted -norm
2010 Mathematics Subject Classification:Primary, 11N45; secondary, 11C08, 60G55, 30C15, 26C10
What is the distribution of algebraic numbers of a given degree ? To answer this question, we first need to define a suitable notion for this distribution. For discrete sets like e.g. infinite subsets of integers one is lead to consider asymptotic relative densities of such sets in intervals of integers in the large limit. Clearly, this direct approach does not work for algebraic numbers (real or complex), since any domain contains infinite number of them (even for a fixed degree ). A possible solution is a classical stratification of algebraic numbers by the concept of height. Let denote the field of (all) algebraic numbers over . A function is called a height function if for any and there are only finitely many algebraic numbers of degree such that . Note that usually it is required (and we will always assume this) that for all conjugates of .
Having defined , we are interested in the asymptotic number of of degree lying in a given subset of or such that as . More generally, for , one would like to determine the asymptotic behaviour of the number of -tuples of conjugate algebraic numbers of degree and height at most as .
In this paper, we consider so-called weighted -heights ().
We would like to emphasize that throughout this paper the degree of polynomials, algebraic numbers, etc. is fixed. We consider algebraic numbers regardless of the particular algebraic number fields they belong to, while counting all vectors with algebraic conjugate coordinates that lie in a given region and have bounded height.
Section 2 contains some basic notation. In Section 3 we describe the problem of counting vectors with algebraic coordinates and formulate the main number-theoretical results of the paper. Section 4 gives a necessary account of related topics in random polynomials. There we formulate the principal result relating distributions of zeros of random polynomials and algebraic numbers. In Section 5 we state several explicit formulae for a function, which plays the role of the joint distribution functions for conjugate algebraic numbers. Sections 6, 7 and 8 contain the proofs of our statements.
2. Basic definitions
Given a polynomial and a vector of positive weights define the -weighted -norm of as
In the non-weighted case , this notion generalizes the naïve height (), the length (), and the Euclidean norm ():
An important weighted example is the Bombieri -norm:
Denote by the -dimensional unit -ball:
Using the well-known formula for the volume of the (non-weighted) unit -ball , it follows that
Let denote the class of integer polynomials (polynomials with integer coefficients) of degree and the -height at most :
We say that an integral polynomial is prime, if it is irreducible over , primitive (the greatest common divisor of its coefficients equals 1), and its leading coefficient is positive. Denote by the class of prime polynomials from :
The minimal polynomial of an algebraic number is a (unique) prime polynomial such that . We put by definition
The other roots of are called (algebraic) conjugates of .
3. Distribution of algebraic numbers
We would like to study the joint distribution of several conjugate algebraic numbers of degree and bounded height. What is the natural configuration space for this problem? Consider a prime polynomial of degree . Some of its zeros are real, and the rest are symmetric with respect to the real line. Thus we may neglect the zeros lying in . Fix some integer numbers such that .
For a Borel subset and the height function denote by the number of ordered mixed -tuples of distinct numbers such that for some it holds
Essentially denotes the number of ordered -tuples in of conjugate algebraic numbers of degree and height at most .
We always assume that , the boundary of , has Lebesgue measure . Our aim is to show that there exists a non-trivial limit
and to find its exact value.
For , a fixed positive vector , and some integer numbers such that there exists a function such that for any Borel we have
where is given in (1).
Provided that is smooth enough we are able to estimate as well the rate of convergence in (2).
Suppose that the boundary of is the union of algebraic surfaces of degree at most . Then,
where depends on only (and may also depend on ) and
It turns out that the function coincides with the correlation function of the roots of some specific random polynomial. To formulate the result we first recall some essential notions.
In case of the naïve height (, ), the problem of finding the limit (2) was solved for real and complex algebraic numbers (i.e. for , , see , and for , , see ), and for vectors with real algebraic conjugate coordinates (, see ).
Unfortunately, our approach cannot cover the case of the Mahler measure. The Mahler measure (in particular, in the form of the Weil height) has many applications in algebraic number theory. Counting algebraic numbers and points with respect to the Weil height and its generalisations has been intensively studied. See the papers , , ,  for results in this direction and related references.
4. Zeros of random polynomials
Let be independent real-valued random variables with probability density functions . Consider the random polynomial defined by
With probability one, all zeros of are simple. Denote by the empirical measure counting the zeros of :
where is the unit point mass at . The random measure may be regarded as a random point process on . A natural way of describing the distribution of a point process is via its correlation functions. Since the coefficients of are real, its zeros are symmetric with respect to the real line, and some of them are real. Therefore, the natural configuration space for the point process must be a ‘‘separated’’ union with topology induced by the union of topologies in and . Instead of considering the correlation functions of the process on , an equivalent way is to investigate the mixed -correlation functions (see ). We call functions , where , mixed -correlation functions of the zeros of , if for any family of mutually disjoint Borel subsets and ,
where denotes the expectation with respect to the (product) distribution of . Here and subsequently, we write
The most intensively studied class of random polynomials are Kac polynomials, when ’s are i.i.d. Sometimes the i.i.d. coefficients are considered with some non-random weights ’s. The common examples are flat or Weil polynomials (with ) and elliptic polynomials (with ).
The -correlation function is called density of real zeros. Integrated over it equals the average number of real zeros of . The asymptotic properties of this object as have been intensively studied for many years, mostly for Kac polynomials; see the historical background in  and the survey of the most recent results in . We just mention some contributions here like: , , , , , , .
Similarly, is called a density of complex zeros being an expectation of the empirical measure counting non-real zeros. Its limit behaviour as is of a great interest as well, see , , , , , , and the references given there.
There are comparatively few papers on higher-order correlation functions of zeros. Well-known results are due to Bleher and Di ,  who studied the correlations between real zeros for elliptic and Kac polynomials, and to Tao and Vu  who proved asymptotic universality for the mixed correlation functions for elliptic, Weil, and Kac polynomials under some moment conditions on .
Our next result connects the limit density of tuples of conjugate algebraic numbers with the correlation function of zeros of the following random polynomial.
Let and let be i.i.d. random variables with a Lebesgue density given by
Consider the random polynomial defined as
Then, the mixed -correlation function of zeros of coincides with the function defined in Theorem 3.1.
The exact formula for is given in Section 5. Let us now formulate one important special case.
4.1. Bombieri 2-Norm
The next theorem shows that the way of counting algebraic numbers with respect to the Bombieri 2-norm is in some sense most natural. It has been shown in  that in the case of Bombieri 2-norm the zeros of the corresponding random polynomial (sometimes called elliptic random polynomial) have a very simple density.
Theorem 4.2 (Edelman–Kostlan ).
Assume that Then
Thus for any degree the asymptotic density of algebraic numbers counted with respect to Bombieri 2-norm coincides with the normalized Cauchy density. In particular, from Theorem 3.1 we have
The volume can be calculated as
5. General Formula for
where we used the following notation for the elementary symmetric polynomials:
It is tacitly assumed that the arguments of ’s are well-defined: we shall restrict ourselves to consider those points only such that all symmetric functions of them are real.
Introduce as well the absolute value of the Vandermonde determinant as:
for all . The following theorem generalises this relation to all mixed -correlation functions.
For all ,
where is defined in (8). Specifically,
Note that the correlations between real zeros and the correlations between complex zeros are essentially given by the same function . In particular, provides a formula for the density of complex zeros as well as for the two-point correlation function of real zeros:
The latter formula (with different notations) was obtained in .
The desired explicit formula for follows by choosing
5.1. Other applications of Theorem 5.1
Taking in (12) yields a formula for the density of real zeros:
where we set .
Taking yields a formula for the density of complex zeros:
where we set .
5.2. -Point Correlation Functions
Taking we obtain the (non-normalized) joint density of all zeros given that has exactly real zeros:
5.3. Correlation Function and Polynomial Discriminant
For an arbitrary monic polynomial define its discriminant as
When , the function gives the correlation between all zeros of when exactly of them are real.
and is the monic polynomial whose zeros are the arguments of :
6.1. Methods of Proof: Counting Integer Points
We reduce counting algebraic numbers to counting corresponding minimal polynomials. So we need to formulate some statements about counting integer points in regions.
For a Borel set denote by the number of points in with integer coordinates, and by the number of points in with coprime integer coordinates:
For a real number and a set denote
Let be an integer. Let be a fixed bounded region. If the boundary has Lebesgue measure 0, then
It is important to note that for the existence of the limit should be necessarily Jordan measurable, since simply Lebesgue measurable isn’t sufficient. For example, if one takes to be the set of points in with rational coordinates, then for any positive integer the set contains integer points, but has Lebesgue measure 0.
The lemma can be easily proved if one considers coverings of by -dimensional cubes with edge . See, for example, [23, Chapter VI §2]. Note that none of the conditions of the lemma can be omitted. ∎
Assuming the conditions of Lemma 6.1 we have
Using the inclusion-exclusion principle (Moebius inversion) one can easily show that
where is any positive number such that .
Then we have
For the first sum it is known that
From Lemma 6.1 we infer that for any there exists such that for all . Hence
Therefore, for any
Thus the lemma is proved. ∎
Lemmas 6.1 and 6.2 provide no estimates for the rate of convergence. Additionally, in these Lemmas the region is kept fixed and therefore cannot depend on . To avoid all these restrictions one needs to restrict oneself to a suitable class of regions. See , , .
Consider a region , , with boundary consisting of algebraic surfaces of degree at most . Then
where depends on only and is defined in (3).
Estimating the number of integer points in a region by its measure is a well-known idea. The most <<ancient>> publication (that relates integer point counting to the volume of a region), which the authors are aware of, is a result by Lipschitz . See as well the classical monograph by Bachmann [1, pp. 436–444] (in particular, formulas (83a) and (83b) on pages 441–442). There are a number of papers generalising such estimates to arbitrary lattices, see e.g.  and .
Let us describe how to calculate .
Given a function and a Borel subset denote by the number of ordered -tuples of distinct numbers such that
For any algebraic number its minimal polynomial is prime, and any prime polynomial is a minimal polynomial for some algebraic number. Therefore we have
On the other hand the right-hand side can obviously be written as
Since , the sum in the right-hand side is finite.
Consider a set (which depends on ) consisting of all points such that
Then, by definition of a primitive polynomial,
Hence it follows from the definition of a prime polynomial that
Note that the factor arises because prime polynomials have positive leading coefficients. It is known (see ) that
Applying this to (16), we obtain
where depends only on the degree , the number of the algebraic surfaces forming , and the maximal degree of these surfaces.
Thus we are left with the task of calculating of . To this end, consider the random polynomial defined as
where the random vector is uniformly distributed over . By definition of ,
The probability on the left-hand side is difficult to calculate due to the dependence of the coefficients of . However, the zeros of do not change if we divide the polynomial by any non-zero constant. By proper normalisation we can achieve independence of the coefficients.
Let , and be a vector of positive weights. Assume that the random vector is uniformly distributed in and that the random variables are i.i.d. with Lebesgue density