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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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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.

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

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 satisfy and then a is a Pisot number. Also, for α∈s it is known that 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 [1].

where then α is a Pisot number. If where and a_{0}≠0 then α is a Pisot number.

If and 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

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- z^{3}- z^{4}- z^{6}- z^{7}+ z^{9}+z^{10}.

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 (α_{1}, α_{2}…, α_{k}) is called a Pisot-k uplet or o(k)-Pisot number if α_{1}, α_{2}…, α_{k} 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. (α_{1}, α_{2})∈S_{2})

**Proposition 1.1:** If (α_{1}, α_{2})∈S_{2})∈S_{2} then (α^{n}_{1} ,α^{n}_{2})∈S_{2} for all .

Proposition 1.2: If (α_{1}, α_{2})∈S_{2} then {α^{n}_{1} ,α^{n}_{2}}→0 for all

Proposition 1.3: If (α_{1}, α_{2})∈S_{2})∈S_{2} with minimal polynomial, of degree 3 and P(0)=1 then α_{1}, α_{2}∈S_{2}

Theorem 1.4: The limit points of S_{2} lie either in S_{2} S× or

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. an algebraic integer over and its minimal polynomial be

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

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

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

where

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

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

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 where and , for all i≠d-1 then Λ is irreducible on .

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

The smallest Pisot number, in the real case, is the only real root of the polynomial x^{3}-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) = Y^{n}-αXY^{n-1}-α^{n} where α is the least element of F_{q}\{0} Moreover, the sequence (w_{n})_{n1} is decreasing one and converge to α_{X} In the present paper we give generalized Pisot and Salem series in the field of formal power series over finite field. P(Y) = Y^{n}-αXY^{n-1}-α^{n}

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 **

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

where

Define the absolute value by

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

and

For define the integer (polynomial) part

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

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

We define the **logarithmic** hauteur of P by

Where log_{q}x designate the logarithmic function in the q basis. For all element algebraic w∈_{q}((X^{-1})) one will note by the logarithmic hauteur of his minimal polynomial.

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

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

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

**Example 2.2**

1) Series of Pisot-2 of degree 2 on

Let

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

For Y_{1} we have

So

Therefore Z_{1} is root of Polynomial irreducible and So

For Y_{2} we have

Then

Therefore Z1 is root of Polynomial irreducible and So

Series of Pisot-2 of degree 3 on

Let

Then P is irreducible over because P and his **reciprocal** polynomial 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 and such that (Y1, Y2)∈2((X-1))2 and one root of P Then

For Y_{1}: we have

Then

Therefore Z_{1} is of type (I) and

For Y_{2}: we have

Then

Thus Z_{3} is of type (I) and

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

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

Proposition 3.1: Let then

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

We know that the product of any two algebraic is, itself, an algebraic. Since w_{1} is an algebraic, then , wn a is also an algebraic. Let be the minimal polynomial of w^{n}_{1} . Now consider the embedding i of into , which fixes and maps w_{1} to w_{i}

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

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

Therefore

**Proposition 3.2:** Let

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

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

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

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

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

Let w_{k+1} be the k+1 root of P Since

Consider

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

**Theorem 3.4:** Let the polynomial

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 . respectively and

Proof. Let w = w_{1},... w_{2}…. w_{n} be the roots of P(Y) such that

We have

Then

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 Then

So

on the other hand

Then

Now, if

Let w = w_{1},... w_{2}…. w_{k} be the roots of P(Y) such that

and

We have

Then

**Consequence 3.5:** Let the polynomial

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 . respectively

**Corollary 3.6:** Let P be the polynomial

such that then Λ has no roots in

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 = w_{1},...w_{2}…. w_{n} be the roots of P(Y) such that

We have

And so

This gives us that

witch is absurd. So Λ is irreducible on

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

If , then Λ is the minimal polynomial of w and (w = w_{1},...w_{k})∈S_{k}

Proof. By the Theorem 3.4, P has exactly k conjugates of w = w_{1},...w_{k} 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 | w_{i} |<1 the authors roots of P. Show that Λ(Y) is irreducible.

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

Let Λ(Y) =Λ_{1}(Y). Λ_{2}(Y) where Λ_{i}(Y), i =1,2, has of the coefficients in

Suppose in the first w_{1},...w_{k} are the roots of Λ_{1} and the other roots are of Λ_{2} Clearly, the absolute value of leading coefficient of the polynomial Λ_{2} superior or equal 1 with is absurd because Λ_{2} has only roots wi such that

0<|w_{i}

|<1

Then we have

This gives us that

If then

And

Also

This gives us that

Therefore

With is absurd.

Now if then

And

This gives us that

We obtain

Therefore

With is absurd.

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

|<1.

We conclude that Λ is the minimal polynomial of and (w_{1}… w_{k})∈S_{k}

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

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

By the consequence 3.5, we have we obtain

Therefore

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

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.

- Brauer A (1951) On algebraic equations with all but one root in the interior of the unit circle. Math Nachr 4: 250-257.
- Akiyama S (2006) Positive finiteness of number systems. Number theory , Tradition and Modernization.
- Bateman P, Duquette AL(1962) The analogue of Pisot-Vijayaraghavan numbers in fields of power series. Ill J Math 6: 594-406.
- Chandoul A, Jellali M, Mkaouar M (2011) Irreducibility criterion over finite fields. Communication in Algebra 39: 3133-3137.
- Chandoul A, Jellali, Mkaouar M (2013) The smallist Pisot element over Fq((X-1)), Communication in Algebra 56: 258-264.
- Sprindzuk VG (1963) Mahler’s problem in metric number theory, Translaion of Mathematical monographs. Amer math Soc.

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