Abstract
The generalisation of questions of the classic arithmetic has long been of interest. One line of questioning, introduced by Car in 1995, inspired by the equidistribution of the sequence where , is the study of the sequence , where is a polynomial having an l-th root in the field of formal power series. In this paper, we consider the sequence , where is a polynomial having an l-th root in the field of formal power series and satisfying . Our main result is to prove the uniform distribution in the Laurent series case. Particularly, we prove the case for irreducible polynomials.
1. Introduction
In 1952, Carlitz [1] introduced the definition of equidistribution modulo 1 in the formal power series case which reveals profitable; it uses Weyl’s criterion [1], the generalisation of van der Corput inequality by Dijksma [2], and the theorem of Koksma by Mathan [3].
Car in [4], inspired by equidistribution modulo 1 of the sequence where , characterised equidistribution modulo 1 of the sequences and , where describes the sequence of polynomials in (resp. describes the sequence of irreducible polynomials in ) with an l-th root (resp. ) in the field of formal power series.
In 2013, Mauduit and Car studied in [5] the automaticity of the set of k-th power of polynomials in . Moreover, they calculated the number of polynomials with degree such that the sum of digits of in base is fixed. In the same subject, Madritsch and Thuswaldner in [6] called the maps , where is the group of Q-additives satisfying for all polynomials with . They proved the equidistribution of the sequence , where is a polynomial with coefficients in the field of formal power series and is an ordered sequence of polynomials in if and only if one of the coefficients of is irrational.
In this article, we are interested in the subsequences of and of of polynomials in arithmetic progression having an l-th root. We will prove that the sequences and are equidistributed modulo 1.
2. Preliminary
Let be a finite field of characteristic with elements. We consider , , and as analogues of , and , respectively.
An element is of the form . We define and . We note the polynomial part of and its fractionary part. Let if , and . Let be a nontrivial additive character. For all , we suppose that .
Let be a positive integer which is not divisible by the characteristic of the field . We introduce as the set of the r-th elements having an l-th root in , and we have
Then, for and , is called an l-th root of ; we note if and only if and . In particular, a nonzero polynomial has an l-th root in if and only if and .
We denote by the set of polynomials with an l-th root in :and if is the set of irreducible polynomials over , we define .
If , where for all , is the representation in base of the integer , then letwhere are given by the bijection from to . For and 1, it is convenient to suppose that and . We define the order in by , for all , and in by
Then, we order by posing for all natural numbers n:
This paper is devoted to the study of equidistribution modulo 1 of a certain sequence in the field of Laurent formal power series. In 1952, Carlitz introduced and characterised equidistribution modulo 1 in the field of Laurent formal power series and obtained the following result.
Lemma 1. (see [1]; Weyl’s criterion). The sequence with values in is equidistributed modulo 1 if and only iffor all .
Finally, we enounce a result which concerns a class of irreducible polynomials given by Artin in [7], which will be very useful later.
Theorem 1. (see [7]). Let be coprime polynomials. If denotes the number of monic irreducible polynomials with degree which are congruent to modulo , thenwhere is a constant . This theorem is analogous to the theorem of prime numbers in arithmetic progression.
3. Results
Let be an integer nondivisible by the characteristic of the field ; we order the set of the l-th powers of under the increasing order of , and we fix a polynomial with degree . For all , we denote by the subset of defined in (2),and the subset of :
We ordered the elements of and with the order relation defined in (2); hence,
The aim of this paper is to prove the following theorems.
Theorem 2. Let be the sequence of polynomials of indexed under the increasing order of . Then, for , the sequence is equidistributed modulo 1.
Theorem 3. Let be the sequence of polynomials of indexed under the increasing order of . Then, the sequence is equidistributed modulo 1 for , and is a constant defined in Theorem 1. In particular, if does not divide C, then let .
4. Proofs of Theorems 2 and 3
4.1. Tools
A generalisation of Theorem 1 was proved in 1965 by Hayes introducing the arithmetic progression.
Lemma 2. (see [8]). Let be a polynomial with degree . Then, for all polynomials , there exist exactly monic polynomials with degree which are congruent to modulo if .
Theorem 4. (see [8]). Let be a positive integer, be a sequence of k elements in , and be coprime polynomials. If, for , is the number of irreducible and monic polynomials with degree which are congruent to modulo such that thenwhere is a constant .
Remark 1. In particular, if does not divide , then (11) is verified for .
The proofs of Theorems 2 and 3 are based on Corollary 1 whose proof needs the following lemmas.
Lemma 3. (see [9], Lemma II.1.1). Let , with degree , and with degree . Then, for all such that , we have(1)(2)
Lemma 4. (see [9], Lemma II.1.2). Let with degree and such that . Then, for all , all , and all with degree , there exists unique such that, for all with degree , we obtain
Corollary 1. Let with degree and such that . For all , we have(i), where (ii), where , where is a constant
Proof. For or , we notewhere is the number of polynomials such that , but with Lemma 4, for with degree , there exists such that the polynomialsatisfyingLet ; we denote by the number of polynomials such thatWe obtain(i)If , then by Lemma 2, we have . With the orthogonality criterion of , it results in(ii)If , then by Theorem 4, we haveWe deduce thatFinally, with the orthogonality criterion, we obtain
4.2. Proof of Theorem 2
In , there are monic polynomials which are congruent to modulo with degree , and let . We denote by (resp. ) the number of polynomials in with degree (resp. ). It is sufficient to verify thatwhere is defined in (1). Let with degree , and is an integer such thatwhere defines the least integer . The sequence is strictly increasing, and there exists a unique integer such that
Moreover, there exists a unique integer , such that
Let
To prove Theorem 2, we have to show that
Using relations (24) and (25), we rewrite the sum to obtainwith
We start by giving an estimation of the sum which concerns the polynomials of with degree . We have
We have to just major the first part of the sum by the number of polynomials with degree , and we apply Corollary 1 on the second part to obtain
We apply the same Corollary 1 on that represents the sum of polynomials with degree lt and and with signature , then
The polynomials in can be written in the formwhere and the sequence is strictly increasing in .
By the order relation on (4), ifand is the presentation in base q of the integer , we have
To estimate , we will distinguish two cases: when the degree of is up to the integer and when it is not: 1st case: . Using (34) and the fact that the sequence is strictly increasing, we obtain Thus, 2nd case: . The polynomials defined in (33) are of the form
Let be the set of polynomials of the formsuch that, for every polynomial with degree , we have
If is the greatest index for which are written in the formwith and being a polynomial with degree , then we rewrite the sum :with
We havewhere is the number of couples such that , with degree , and . Moreover, we havewhere denotes the number of polynomials such thatwithwhich gives the same arguments presented in the proof of Corollary 1, and then we deduce that
In (34), let be the least index such that ; then, we have
Since all polynomial of the formwhich coefficients satisfy the condition:is less then , we obtainwhich leads to
Then, with (48) and (53), it results in
With (31), (32), and (54), we obtainand finally, with (24), we obtainwhich ends the proof.
4.3. Proof of Theorem 3
The proof of Theorem 3 is treated as the proof of Theorem 2, and we will keep the same notations with the appropriate modifications. Let be the number of monic irreducible polynomials with degree in , congruent to modulo , satisfying with [4] the following property:
In , we have , and there exist constants and such that
Let be a nonzero polynomial with degree and be an integer satisfying (23). We suppose now that
With relations (24) and (25), we obtain
By the same method used in the proof of Theorem 2, with Corollary 1, we obtainand thenwhere is the sum defined in (42) concerning the polynomials in . Finally, we treat the sum
With Theorem 4, for all polynomialsin which coefficients satisfy condition (51), there exists
Irreducible polynomials are congruent to modulo such that Such polynomials are in , and we have
Then,
Finally, from (61), (62), and (69), we have
Then,which gives the corresponding conclusion needed for .
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares that there are no conflicts of interest.
Acknowledgments
The author gratefully acknowledges the approval and the support of this research study from the Deanship of Scientific Research at Northern Border University, Arar, KSA (Grant no. SCI-2017-1-8-F-7504).