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ISSN: 2090-0902
Journal of Physical Mathematics
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Pisot-K Elements in the Field of Formal Power Series over Finite Field

Kthiri H*

Department of Mathematics, University of Sfax, Tunisa

*Corresponding Author:
Kthiri H
Department of Mathematics, University of Sfax, Tunisa
Tel: +216 74 242 951
E-mail: [email protected]

Received Date: June 15, 2016; Accepted Date: August 22, 2016; Published Date: August 25, 2016

Citation: Kthiri H (2016) Pisot-K Elements in the Field of Formal Power Series over Finite Field. J Phys Math 7: 194. doi: 10.4172/2090-0902.1000194

Copyright: © 2016 Kthiri H. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

In this paper, we will give a criteria of irreducible polynomials over q[X] where q is a finite field. We will present an estimation for the number of the Pisot-k power formal series, precisely we will give their degrees and their logarithmic heights

Keywords

Formal power series; Irreducible polynomials; Pisot-2 series. |z|<1

Introduction

In 2007, Gabriel Ponce, Melanie Ruiz, Emily McLeod Schnitger and Noah Simon have discuss certain sets of algebraic integers related to Pisot numbers (An integer algebraic number α>1 is called a Pisot number if all its conjugates, different from α lie in the open disc |z|<1 on the complex plane ) and Salem number (An integer algebraic number α>1 is called a Salem number if all its conjugates, different from α lie in the disc |z|<1 and it has at least one conjugate having modulus equal to 1) Also their properties.

Much is known about Pisot numbers, for example, if the integer coefficients of the minimal polynomial of image satisfy image and image then a is a Pisot number. Also, for α∈s it is known that image One important theorem about Pisot numbers is that the set S is closed and there are many known ways to construct them.

Let α be a root dominate of polynomial image [1].

image where image then α is a Pisot number. If image where image and a0≠0 then α is a Pisot number.

If image and image then α is a Pisot number [2].

In comparison, little is known about Salem numbers, and their construction is difficult. There are still many open questions about Salem numbers, including determining the infest of the set.

Definition 1.1: The reciprocal polynomial of a polynomial P is defined by

image A polynomial is called reciprocal if P=P* In general, we will denote the reciprocal polynomial of P by Q.

The minimal polynomial of a Salem number will always be reciprocal, and thus, there are no Salem numbers with odd degree or degree less than 4. Constructing Salem numbers is much more difficult than Pisot numbers. For instance, graph theory can be used to construct some but not nearly all. The smallest Salem number is still not known, though it is conjectured to be largest root of 1+z- z3- z4- z6- z7+ z9+z10.

They have defined new sets of generalized Pisot numbers and they have concerned with the arithmetic properties and limit points of one of these sets, Pisot-2 pairs (A pair of real distinct algebraic conjugates (α1, α2…, αk) is called a Pisot-k uplet or o(k)-Pisot number if α1, α2…, αk are greater than 1 and all remaining conjugates have modulus strictly less than 1) We denote the set of Pisot-k pairs by Sk For this set they have obtained some results analogous to those known about Pisot numbers. (α1, α2)∈S2)

Proposition 1.1: If (α1, α2)∈S2)∈S2 then (αn1n2)∈S2 for all image.

Proposition 1.2: If (α1, α2)∈S2 then {αn1n2}→0 for all image

Proposition 1.3: If (α1, α2)∈S2)∈S2 with minimal polynomial, image of degree 3 and P(0)=1 then α1, α2∈S2

Theorem 1.4: The limit points of S2 lie either in S2 S× or image

A great deal is known about this set. Then, they have discussed another set, Pisot (o)-2 numbers, and its connections to Salem numbers, including a relationship with the infemum of Salem numbers. Finally, They have giving arithmetic properties of these Pisot (o)-2 numbers. In this paper, we consider an analogue of this concept in algebraic function over finite fields.

Recall that in 1962 Batemen and Duquettes introduced and characterized the elements of Pisot and Salem in the field of formal power series [3].

Theorem 1.5: [4] Let. image an algebraic integer over image and its minimal polynomial be

image

Then w is a Pisot (respectively Salem) series if and only if image image

Chandoul, Jellali and Mkaouar have improve the Theorem of Bateman and Duquette [4] on image and this while establishing the same result with weaker hypotheses [4].

Theorem 1.6: [2] Let image be Pisot (respectively Salem) series if and only if w is a root of polynomial

image

where imageimage

In the same setting and on the field of the real, Brauer, gave a criteria of irreducibility on image [5]

Theorem 1.7: If image are of the integer such that image, then the polynomial image is irreducible on image

Chandoul, Jellali and Mkaouar are constructs the analog of the theorem of Brauer in the case of the polynomials to coefficients in q [X] [4].

Theorem 1.8: Let image where image and image, for all i≠d-1 then Λ is irreducible on image.

This last theorem permits to give a new evaluation to calculate the number of elements of Pisot on image being given their n degrees and their logarithmic heights h This evaluation is illustrated in the following result:

image

The smallest Pisot number, in the real case, is the only real root of the polynomial x3-x-1 known as the plastic number or number of money (approximately 1,324718 But, as we have said, the smallest accumulation point of all Pisot numbers is the golden number. Chandoul, Jellali and Mkaouar are prove in [5] that the minimal polynomial of the smallest Pisot element (SPE) of degree n is P(Y) = Yn-αXYn-1n where α is the least element of Fq\{0} Moreover, the sequence (wn)n1 is decreasing one and converge to αX In the present paper we give generalized Pisot and Salem series in the field of formal power series over finite field. P(Y) = Yn-αXYn-1n

The paper is organized as follows. In this section, we give some preliminary definitions, we define Pisot numbers and Salem numbers. We give the well known properties of its. In section 2 we introduce the field of formal power series over finite field, we define a totally order, the lexicographic order, over a field of formal power series with coefficients in a finite field. In section 3 We give the arithmetic Properties of Pisot (o)-k series and the criteria of irreducible of this series.

of Formal Series image

Let be image a finite field of q elements, image the ring of polynomials with coefficient image the field of rational functions, image the minimal extension of image containing X and β and image the minimal ring containing X and β. Let image be the field of formal power series of the form :

image

where

image

Define the absolute value by

image

Then |. | is not archimedean. It fulfills the strict triangular inequality

image

and

image

For image define the integer (polynomial) part image

where the empty sum, as usual, is defined to be zero. Therefore image and (w-[w]) is in the unit disk D(0,1) for all image. As explained by Sprindzuk a non archimedean absolute value on image is defined by |w|= q-s It is clear that, for all image and, for all image [X];such that image [6].

We know that image is complete and locally compact with respect to the metric defined by this absolute value. We denote by image an algebraic closure of image We note that the absolute value has a unique extension to image. Abusing a little the notations, we will use the same symbol |.| for the two absolute values. For all polynomial (P≠0)

image

We define the logarithmic hauteur image of P by

image

Where logqx designate the logarithmic function in the q basis. For all element algebraic w∈q((X-1)) one will note by image the logarithmic hauteur of his minimal polynomial.

Theorem 2.1: Let image is of Pisot number if and only if it exist image such that image Moreover λ can be chosen to belong to image.

Recall that image contains Pisot elements of any degree over image Indeed, consider the polynomial Yn-aYn-1-b where image it can be seen easily, considering its Newton polygon, that the polynomial, which is irreducible over q(X) has a root image such that |w|>1 and all of its conjugates in image have an absolute value strictly smaller than 1.

Definition 2.1: An uplet of series algebraic conjugates(w1…wk)image is called a series of Pisot-k uplet (o(k)-Pisot series ) if w1 integer algebraic such that |w1|,….|wk| are greater than 1 and all remaining conjugates have modulus strictly less than 1 We denote the set of Pisot-k uplet by image.

Example 2.2

1) Series of Pisot-2 of degree 2 on image

Let

image

Then P is irreducible over image Now we show that P has two roots image such that image Let image one root of P.Then

image

For Y1 we have

image

So

image

Therefore Z1 is root of Polynomial irreducible and image So image

For Y2 we have

image

Then

image

Therefore Z1 is root of Polynomial irreducible and image So image

Series of Pisot-2 of degree 3 on image

Let image

Then P is irreducible over image because P and his reciprocal polynomial image are the same natures. However Q is of type (I) then it is irreducible. Therefore P is also.

Now we show that P has two roots image and image such that (Y1, Y2)∈image2((X-1))2 and image one root of P Then image

image

For Y1: we have

image

Then

image

Therefore Z1 is of type (I) and image

For Y2: we have

image

Then

image

image

image

image

image

Thus Z3 is of type (I) and image

Results

Arithmetic properties of Pisot-k uplet (respectively Salem) series

In this section we discuss some basic arithmetic properties of image. It is known that image then image Also, we have that image where {x} denotes the fractional part of x. Our last proposition relates a specific subset of image to S′ The algebraic closure of image will be denoted by image.

Proposition 3.1: Let image then image

Proof. Let image (respectively. image) and image the minimal polynomial of w and w = w1,... wd the conjugates of w. Then there exist exactly k conjugates w = w1,... wk of w that lie outside the unit disc. Let wk+1…wd denote the other roots of M.

We know that the product of any two algebraic is, itself, an algebraic. Since w1 is an algebraic, then image, wn a is also an algebraic. Let image be the minimal polynomial of wn1 . Now consider the embedding i of image into image, which fixes image and maps w1 to wi

image

So for all image satisfies P(Y)=0 We have, image. This shows that deg(P)≤deg(M) So image are all the roots of P

If image then image (respectively. there are at least k+1≤j≤d such that image and image

Therefore image

Proposition 3.2: Let image

Proof. Let w1 be an Pisot series and w1.. wd its conjugates. By the proof of theorem 3.1, for all image are the roots of some degree d irreducible polynomial, pn in image Also,

image

So {tr(Pn)}=0 The above can be rewritten as

image

Since, for k+1≤i≤d by definition image therefore image. Thus image. Therefore image

Proposition 3.3: Let image with minimal polynomial image of degree k+1 and w = w1,… wk... wk+1 the conjugates of w If w is unit then image

Proof. Let image with minimal polynomial P has degree k+1 and P(0)=C

Let wk+1 be the k+1 root of P Since

image

Consider

image

Clearly Q is unit, irreducible over image and has roots image We have image and image Therefore image is a Pisot series.

Formal Pisot-k pairs (respectively Salem) Series

Theorem 3.4: Let the polynomial

image

P has exactly k roots that lie outside the unit disc sauch that one the these roots are the biggest and all remaining roots have modulus strictly less than 1(respectively the other roots are have modulus inferior or equal to 1 and at least exist a root of module equals to 1) if and only if image. respectively image and image

Proof. Let w = w1,... w2…. wn be the roots of P(Y) such that image

We have

image

Then

image

Second, Prove the sufficiency by the symmetrical relations of the roots of a polynomial. Let w = w1,... w2…. wn be the roots of P(Y) such that image Then

image

So

image

on the other hand

image

Then

image

Now, if image

Let w = w1,... w2…. wk be the roots of P(Y) such that

image and image

We have

image

Then

image

Consequence 3.5: Let the polynomial

image

P has exactly k roots that lie outside the unit disc sauch that one the these roots are the biggest and all remaining roots have modulus strictly less than 1(respectively the other roots are have modulus inferior or equal to 1 and at least exist a root of module equals to 1) if and only if image. respectively image

Corollary 3.6: Let P be the polynomial

image

such that image then Λ has no roots in image

Proof. By the previous Theorem Λ has k roots of modules >1 with a different value and the other roots are of modules <1 Let w = w1,...w2…. wn be the roots of P(Y) such that

image

We have

image

And so

image

This gives us that

image

witch is absurd. So Λ is irreducible on image

Theorem 3.7: Let image witch image be the are roots of the polynomial

image

If image, then Λ is the minimal polynomial of w and (w = w1,...wk)∈Sk

Proof. By the Theorem 3.4, P has exactly k conjugates of w = w1,...wk that lie outside the unit disc and all remaining conjugates have modulus strictly less than <1 Let’s wi for i=k+1,…n witch | wi |<1 the authors roots of P. Show that Λ(Y) is irreducible.

By the condition of the Theorem, Λ(0)=λ0≠0 hence, all roots of the polynomial Λ(Y). are not equal to 0

Let Λ(Y) =Λ1(Y). Λ2(Y) where Λi(Y), i =1,2, has of the coefficients in image

Suppose in the first w1,...wk are the roots of Λ1 and the other roots are of Λ2 Clearly, the absolute value of leading coefficient of the polynomial Λ2 superior or equal 1 with is absurd because Λ2 has only roots wi such that

0<|wi

|<1

 

image

Then we have

image

This gives us that

image

If image then

image

And

image

image

Also

image

image

This gives us that

image

Therefore

image

With is absurd.

Now if image then

image

And

image

image

This gives us that

image

image

We obtain

image

Therefore

image

With is absurd. image

Let Λ(Y)=where Λi(Y),i=1,2,3 has of the coefficients in image. Suppose in the first w1,...wk are the roots of image are the roots of Λ2, such that image and the other roots are of Λ3. Clearly, the absolute value of leading coefficient of the polynomial Λ3 superior or equal 1 with is absurd because Λ2 has only roots wi such that 0<|wi

|<1.

 

We conclude that Λ is the minimal polynomial of and (w1… wk)∈Sk

Corollary 3.8: Let image the set of the element of Pisot-k series in image of degree n and of height logarithmic h Then the numbers of the elements of image are

image

Proof. The set of the minimal polynomials of the elements of image (n,h) is

image

image

By the consequence 3.5, we have image we obtain

image

Therefore

image

Conclusion

The presentation and estimation for the number of the Pisot-k power formal series, precisely we will give their degrees and their logarithmic heights.

Acknowledgements

I would like to thank Dr. Mabrouk ben ammar and Amara chandoul for all of his guidance and instruction, and for introducing me to the study of Pisot numbers.

References

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