Pisot-K Elements in the Field of Formal Power Series Over Finite Field Kthiri H* Department of Mathematics, University of Sfax, Tunisa

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o s J Journal of Physical Mathematics DOI: 10.4172/2090-0902.1000194 ISSN: 2090-0902

Research Article Article OpenOpen Access Access Pisot-K Elements in the Field of Formal Power Series over Finite Field Kthiri H* Department of Mathematics, University of Sfax, Tunisa

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.

Formal power series; Irreducible polynomials; Pisot-2 nn Keywords: Proposition 1.2: If (α1, α2) S2 then {αα12+→ }0 for all n  series. |z|<1 Proposition 1.3: If (α1, α2)∈S2 with minimal polynomial, P∈ [X] of Introduction degree 3 and P(0)=1 then α1, α2 S2 ∈ ∈ In 2007, Gabriel Ponce, Melanie Ruiz, Emily McLeod Schnitger Theorem 1.4: The limit points∈ of 2S lie either in S2 S× or × S and Noah Simon have discuss certain sets of algebraic integers related A great deal is known about this set. Then, they have discussed to Pisot numbers (An integer algebraic number α>1 is called a Pisot another set, Pisot (o)-2 numbers, and its connections to Salem numbers, number if all its conjugates, different from α lie in the open disc |z|<1 on ) including a relationship with the infemum of Salem numbers. Finally, the complex plane and Salem number (An integer algebraic number They have giving arithmetic properties of these Pisot (o)-2 numbers. α>1 is called a Salem number if all its conjugates, different from α lie in In this paper, we consider an analogue of this concept in algebraic the disc |z|<1 and it has at least one conjugate having modulus equal to function over finite fields. 1) Also their properties. Recall that in 1962 Batemen and Duquettes introduced and Much is known about Pisot numbers, for example, if the integer characterized the elements of Pisot and Salem in the field of formal coefficients of the minimal polynomial of α Pz( )= zdd+ q z−1 ++ q, d−1 d−10power series [3]. 1+ q <0 dd−1 satisfy ∑ i=0 i and Px( )= x− ad−10 x −− a. then a is a Pisot n -1 number. Also, for α s it is known that α s One important theorem Theorem 1.5: [4] Let. w q((X )) an algebraic integer over q [X] about Pisot numbers is that the set S is closed and there are many and its minimal polynomial be ∈ known ways to construct∈ them. ∈ nn−1 Py( )= y+ An−10 y ++ A,[] Aiq ∈ X dd−1 Let α be a root dominate of polynomial Px( )= x− a− x −− a.[1]. d 10 Then w is a Pisot (respectively Salem) series if and only if * |AA |> | | ( |AA |=in≠−1 | |). aadd−−12≥ ≥≥ a 0>0 where a then α is a Pisot number. If ni−1 max respectively 1 max i d−2 i  in≠−1 aa> | |1+ di−1 ∑ i=0 where ai  and a0≠0 then α is a Pisot number. d ∈ Chandoul, Jellali and Mkaouar have improve the Theorem of -1 aa− > Bateman and Duquette [4] on ((X )) and this while establishing the If di1 ∑ ,ai≥0 and a∈d≠0, then α is a Pisot number [2]. q i=2 same result with weaker hypotheses [4]. In comparison, little is known about Salem numbers, and their -1 construction is difficult. There are still many open questions about Theorem 1.6: [2] Let w q((X )) be Pisot (respectively Salem) series Salem numbers, including determining the infest of the set. if and only if w is a root of polynomial ∈ nn−1 Definition 1.1: The reciprocal polynomial of a polynomial P is Py( )= y+ An−10 y ++ A,[] Aiq ∈ X ∗ 1 Pz( ) = zdeg P P ( ). * defined by A polynomial is called reciprocal if P=P In where |AAni−1 |>max | | ( respectively |AAni−1 |=max in≠−1 | |). z in≠−1 general, we will denote the reciprocal polynomial of P by Q. In the same setting and on the field of the real, Brauer, gave a The minimal polynomial of a Salem number will always be criteria of irreducibility on  [5] reciprocal, and thus, there are no Salem numbers with odd degree or Theorem 1.7: If aa12,,, an− 1 are of the integer such that degree less than 4. Constructing Salem numbers is much more difficult nn−−12 n aa−−≥ ≥≥ a> 0, then the polynomial x− ax−− − ax −− a than Pisot numbers. For instance, graph theory can be used to construct nn12 0 nn12 0 is irreducible on  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 *Corresponding author: Kthiri H, Department of Mathematics, University of Sfax, of these sets, Pisot-2 pairs (A pair of real distinct algebraic conjugates Tunisa, Tel: +216 74 242 951; E-mail: [email protected]

(α1, α2…, αk) is called a Pisot-k uplet or o(k)-Pisot number if α1, α2…, αk Received June 15, 2016; Accepted August 22, 2016; Published August 25, 2016 are greater than 1 and all remaining conjugates have modulus strictly Citation: Kthiri H (2016) Pisot-K Elements in the Field of Formal Power Series over less than 1) We denote the set of Pisot-k pairs by Sk For this set they have Finite Field. J Phys Math 7: 194. doi: 10.4172/2090-0902.1000194 obtained some results analogous to those known about Pisot numbers. Copyright: © 2016 Kthiri H. This is an open-access article distributed under the (α1, α2) S2 terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and ααnn∈ Proposition∈ 1.1: If (α1, α2) S2 then (,)12S 2 for all n . source are credited. ∈ ∈

J Phys Math, an open access journal Volume 7 • Issue 3 • 1000194 ISSN: 2090-0902 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

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Chandoul, Jellali and Mkaouar are constructs the analog of the -s degP by |w|= q It is clear that, for all P q[X],|P|=q and, for all Q q theorem of Brauer in the case of the polynomials to coefficients in P − q [X];such that Q≠0 | |=qdegP degQ .[6]. [X] [4]. Q ∈ ∈ − Λdd +λλ1 ++ -1 Theorem 1.8:Let (YY )= d−10 Y where λλiq∈≠[X ],0 0  We know that q((X )) is complete and locally compact with λλ and degdi−1 > deg , for all i≠d-1 then Λ is irreducible on q [X]. respect to the metric defined by this absolute value. We denote by ((X −1 )) an algebraic closure of ((X-1)) We note that the absolute This last theorem permits to give a new evaluation to calculate the q q value has a unique extension to ((X −1 )). Abusing a little the notations, number of elements of Pisot on ((X-1)) being given their n degrees q q we will use the same symbol |.| for the two absolute values. For all and their logarithmic heights h This evaluation is illustrated in the polynomial (P≠0) following result: P( Y )= Ynn+ AY−1 ++ A,[] A ∈ X (nh , ) = ( q−− 1)( qh 1) q( nh− 1) . 1 ni q We define the logarithmic hauteur (P) of P by The smallest Pisot number, in the real case, is the only real root of  3 the polynomial x -x-1 known as the plastic number or number of money ()=Plog qmax | AAii |= max deg() 00≤≤in ≤≤in (approximately 1,324718 But, as we have said, the smallest accumulation point of all Pisot numbers is the golden number. Chandoul, Jellali and Where log qx designate the logarithmic function in the q basis. For -1 Mkaouar are prove in [5] that the minimal polynomial of the smallest all element algebraic w q((X )) one will note by (w) the logarithmic Pisot element (SPE) of degree n is P(Y) = Yn-αXYn-1-αn where α is the hauteur of his minimal polynomial. ∈ least element of F \{0} Moreover, the sequence (w ) is decreasing one -1 q n n1 Theorem 2.1: Let w q((X )) is of Pisot number if and only if it exist -1 n and converge to αX In the present paper we give generalized Pisot and →+∞{λw }=0 λ q((X ))\{0} such that limn ; Moreover λ can be chosen Salem series in the field of formal power series over finite field. P(Y) = ∈ to belong to q(X)(w). Yn-αXYn-1-αn ∈ -1 Recall that q((X )) contains Pisot elements of any degree over The paper is organized as follows. In this section, we give some n n-1 q(X) Indeed, consider the polynomial Y -aY -b where a,b q\{0} it preliminary definitions, we define Pisot numbers and Salem numbers. can be seen easily, considering its Newton polygon, that the polynomial, We give the well known properties of its. In section 2 we introduce the -1 ∈ which is irreducible over q(X) has a root w q((X )) such that |w|>1 field of formal power series over finite field, we define a totally order, the −1 and all of its conjugates in q ((X )) have an absolute value strictly lexicographic order, over a field of formal power series with coefficients smaller than 1. ∈ 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. Definition 2.1: An uplet of series algebraic conjugates(w1… -1 k ) wk)q((X ))) is called a series of Pisot-k uplet (o(k)-Pisot series -1 of Formal Series q((X )) 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 Let be q a finite field of q elements, q [X] the ring of polynomials of Pisot-k uplet by Sk′. with coefficient qI , q(X) the field of rational functions, q(X, β) the minimal extension of q containing X and β and q(x, β) the minimal Example 2.2 ring containing X and β. Let ((X-1)) be the field of formal power series q 1) Series of Pisot-2 of degree 2 on ((X-1)) (w … w ) ((X-1)))k of the form : 2 1 k q l Let w=, wXk w∈ ∑ k kq 22 k=−∞ PY( ) = Y++ Y X + X +1. Then P is irreducible over [X][Y] Now we show that P has two roots where 2 ∞ -1 2 Y= YX−−i ∈ (( X 1 )) |Y |>1|Y |>1 such that (Y , Y )  ((X )) Let ∑ i=1− i 2 max{k : w≠≠ 0} for w 0 1 2 1 2 2  k one root of P.Then l= deg w := −∞ for w = 0. ∈   11 ++ + ++  YX11=12 =1X such that ||>2; Z   XZ1  Define the absolute value by  11 YXX22= ++ = + such that| Z |> 1. deg w  XZ q for w ≠ 0  2  |w |= 0 for w = 0. For Y1 we have  22++ ++  Y11 YX X1 = 0. Then |. | is not archimedean. It fulfills the strict triangular inequality So

2 |w+≤ v | max (| w |,| v |) and ZZ11++1 = 0.

-1 Therefore Z1 is root of Polynomial irreducible and Z1 2((X )) So |w+≠ v |= max (| w |,| v |) if | w | | v |. -1 l Y1 2((X )) -1 [w ]= wXk ∈ For f q((X )) define the integer (polynomial) part ∑ k k=0 ∈For Y2 we have where the empty sum, as usual, is defined to be zero. Therefore f [X] 32 ∈ q ++ ++ -1 YYX22 X1 = 0. and (w-[w]) is in the unit disk D(0,1) for all w q((X )). As explained -1 ∈ by Sprindzuk a non archimedean absolute value on q((X )) is defined Then ∈

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2 n ZZ22++1 = 0. Since w1 is an algebraic, then n  ∀∈nw, 1 a is also an algebraic. n w1 . -1 Let P q [X][Y] be the minimal polynomial of Now consider the Therefore Z is root of Polynomial irreducible and Z  ((X )) So ∀ ∈ −1 1 2 2 embedding of  (X) w into q ((X )), which fixes (X) and maps Y ((X-1)) i q 1 q 1 2 w to ∈w ∈ 1 i -1 Series of Pisot-2 of degree 3 on 2((X )) n nnn ∈ Pw(ii ) = P ((σ ( w1 ) )) = P ( σσ i ( w 11 )) = i ( Pw ( )) = σ i (0) = 0 32 23 Let PY( ) = Y++ ( X 1) Y ++ ( X X ) Y + 1. n So for all i≤d wi satisfies P(Y)=0 We have, [ (Xw )(n ) : ( X )]≤ [ ( Xw )( ) : ( X )]. This shows that deg(P)≤deg(M) Then P is irreducible over 2[X][Y] because P and his reciprocal q11  qq  q 33 22 nn n polynomial QY( ) = Y+ ( X + X ) Y + ( X ++ 1) Y 1 are the same natures. So ww12, ,, wd are all the roots of P

However Q is of type (I) then it is irreducible. Therefore P is also. nn If k+1≤i≤d then |wwii |=| | < 1(respectively. there are at least nn nn Now we show that P has two roots |Y |>1 and |Y |>1 such that (Y k+1≤j≤d such that |wwjj |=| | = 1) and |wwii |=| | > 1 for I = 1….k ∞ 1 2 1, Y ) ((X-1))2 and |Y |<1 Let Y= YX−−i ∈ (( X 1 )) one root of P 2 2 3 ∑ i=2− i 2 nn ∈ ′ ∈ ′ * -1 2 Therefore (,,w1 wSkk ) (respectively. Tk ) for all n Then (Y1, Y2) 2((X )) ∈ nn Proposition 3.2: Let (,,w1  wSkk )∈ ′ then {ww1 +→ ,}k 0 as n→∞  22∈ 11 YXX= ++ + = X ++ X such that| Z |> 2;  112 Proof. Let w be an Pisot series and w .. w its conjugates. By the  XZ1 1 1 d wwnn,,  11 proof of theorem 3.1, for all n  a 1 k are the roots of some YX22=1++ + =1X ++ such that||>1; Z XZ degree d irreducible polynomial, Pn in q [X] Also,  2 d ∈  11 n tr( Pn )= w iq∈ [ X ] Y33=+ = such that| Z |> 1. ∑  i=1  XZ3

So {tr(Pn)}=0 The above can be rewritten as For Y1: we have dd Y32++( X 1) Y 23 ++ ( X XY ) + 1 = 0. nn n n 11 1 {tr ( Pni )=∑∑ w }={ w1 ++ wki + w } i=1 ik=+ 1 Then n → Since, for k+1≤i≤d by definition w| i|<1 therefore wi 0. Thus d 3 432 2 n → nn Z1+( XXXXZX +++)11 ++ ( 1) Z + 1 = 0. 0. +→ {}∑ ik=1+ wi Therefore {ww1 k } 0. Therefore Z is of type (I) and Z ((X-1)) So Y ((X-1)) 1 1 2 1 2 Proposition 3.3: Let (,,w1  wSkk )∈ ′ with minimal polynomial p [X][Y] of degree k+1 and w = w ,… w ... w the conjugates of w If For Y2: we have ∈ ∈ q 1 k k+1 k+1 32 23 (− 1) Y++( X 1) Y ++ ( X XY ) + 1 = 0. w is unit then wwS1  k ∈ ′. 22 2 ∈ c  ∈ ′ Then Proof. Let (,,w1 wSkk ) with minimal polynomial P has degree

33 22 k+1 and P(0)=C Z1++( X XZ ) 11 +++ ( X 1) Z 1 = 0.

()I -1 -1 Let wk+1 be the k+1 root of P Since So Z2 is of type and Z2 2((X )) So Y22((X )) 1 PYYwYw( ) = (−−12 )( ) ( Yw −k+ 1 ). For Y :we have Y3 = such that |Z |>1 Then 3 Z ∈ 3 3 Consider Y32++( X 1) Y 23 ++ ( X XY ) + 1 = 0. 33 3 1 1 1 (− 1)k+1 33 22 , ,, = ww. Z++( X XZ ) ++ ( X 1) Z + 1 = 0. 1 k 3 33 ww12 wk+ 1 c -1 -1 Thus Z3 is of type (I) and Z32((X )) So Y3 2((X )) Clearly Q is unit, irreducible over q[X][Y]. and has roots k+1 1 1 1 (− 1)k+1 1 (− 1) ww  1 k Results ∈ , ,, = ww1 k . We have | |=| |=|ww1  k |> 1 ww w+ c 12k 1 wck+1 1 Arithmetic properties of Pisot-k uplet (respectively Salem) | |< 1 k+1 and for i=1, 2,….k Therefore (− 1) ww1  k is a Pisot series. series wi Formal Pisot-k pairs (respectively Salem) Series In this section we discuss some basic arithmetic properties of Sk′. n n It is known that (α,β) S2 if (α,β) S2 then (α ,β ) S2 n  Also, we nn Theorem 3.4: Let the polynomial have that limn→∞{αβ , }→ {0}, where {x} denotes the fractional part of ∈ ∈ ∈′ ∀ ∈ nn−−12 n x. Our last proposition relates a specific subset of Sk to S The algebraic P( Y ) = AYn+ A n−−1 Y + A n 2 Y ++ AY1 + A 00, A ≠ 0, Aiq ∈ [ X ]. -1 −1 closure of q((X )) will be denoted by q ((X )). ′ P has exactly k roots that lie outside the unit disc sauch that one (,,w wS )∈ ′ ( ∈T′), Proposition 3.1: Let 1 kkrespectively. k then the these roots are the biggest and all remaining roots have modulus (,,wnn wS )∈ ′ ( ∈T′ * 1 kkrespectively. k ) for all n . 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) Proof. Let (,,w1 wSkk ) (respectively. Tk ) and M q [X][Y] the minimal polynomial of w and w = w1,... wd the conjugates of w. Then if and only if |AAnk− |>supink≠− |i |. respectively |AAnk− |>supink≠− |i | and there exist exactly k conjugates w = w ,... w of w that lie outside∈ the unit 1 k |AAnk− |=supink< − |i |.) disc. Let wk+1…wd denote the other roots of M. Proof. Let w = w ,... w …. w be the roots of P(Y) such that We know that the product of any two algebraic is, itself, an algebraic. 1 2 n

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Then |ww12 |>| |≥≥ | wkk |>1>| w+1 | ≥≥ | w n |.

|Ank−− |>sup | Ai | and | Ank |=sup | Ai |. We have ink><−−ink  AAConsequence 3.5: Let the polynomial |w |=|n−−1 |≤ | w | <|ww |=|nk |  ∑ ik11 nn−−12 n ≤≤ AA+ + ++ + ≠ ∈  1 in nnP( Y ) = AYn A n−−1 Y A n 2 Y AY1 A 00, A 0, Aiq [ X ].  AA |w w |=|n−−2 |≤ | ww | <|w w |=|nk | P has exactly k roots that lie outside the unit disc sauch that one  ∑ ii12 12 1 k ≤≤ AA  1| nk −+11 |> >| An− |>supink< − | Ai |.  1supink≠− |i | and nk− supink< − i  Ank− −−1 Ank | ww w |=| |≤ |ww+ | <| ww |=| | ∑ ii12 ik+ 1 11kk 1  1<≤ii ≤≤ i ≤ n AAnn  12k+ 1 Corollary 3.6: Let P be the polynomial nn−−12 n  Λ(YY ) = λ + λ Y + λ Y ++ λ Y + λλ, ≠ 0, λ ∈ [X ]  n n−−1 n 2 1 00 iq AA  0 nk− |∏win |=| |=| ww | <|wwk |=| | such that |λλ− |>sup | |. If |λλλ− |≥ | −+ || − |, then Λ has no roots  1≤≤in AAnnnk ink≠− i nk nk11 n  in ((X-1))  q Then Proof. By the previous Theorem Λ has k roots of modules >1 with a

different value and the other roots are of modules <1 Letw = w1,...w2…. |AAnk− |>sup |i |. ink≠− wn be the roots of P(Y) such that Second, Prove the sufficiency by the symmetrical relations of the |ww12 |>| |≥≥ | wkk |>1>| w+1 | ≥≥ | w n |. roots of a polynomial. Let w = w1,... w2…. wn be the roots of P(Y) such that |www123 |>| |≥ | | ≥≥ | wn |. Then We have λλλ≥ AAnk−+1 nk− |nk− | | nk −+11 || n − |. | |=| www|=| ww− |<| |=| ww  |. ∑ ii12 ik− 1 11k 1k AA≤ ≤≤ ≤ nn1 1 and | w |>| w |≥ | w | ≥≥ | w |> 1. kk123 This gives us that on the other hand |wwk |≥ |1 |.

AAnk−−1 nk− | |=| www|=| ww+ |<| |=| ww  |. witch is absurd. So Λ is irreducible on [X][Y] ∑ ii12 ik+ 1 11k 1k q AAnn1<≤ii ≤≤ i ≤ n 12k+ 1 −1 Theorem 3.7: Let (wXiq )∈  (( )) for ik= 1, , witch ≥ Then |ww12 |>| | | wk |> 1 be the are roots of the polynomial Λnn +λ −−12 + n ++ λ + λλ ∈ λ ≠ λ λ |w+ |<1 and 1>| w++ |>| w |≥≥ | w |. (YYYAYY ) = n−−1 n 2 10,iq [X ],0 0 and |n−2 |>sup | i | k1 kk12 n in≠−2

Now, if |AAnk− |=supink< − |i |. If degλλnk− =degnk−+11+ deg λn− , then Λ is the minimal polynomial of

w and (w = w1,...wk) Sk Let w = w1,... w2…. wk be the roots of P(Y) such that Proof. By the Theorem 3.4,P has exactly k conjugates of w = w ,...w |w |>| w |≥≥ | w |> 1 >| w+ | ≥≥ | w | and ∃+≤≤ k 1 j n such that| w |= 1. ∈ 1 k 12 kk1 n j that lie outside the unit disc and all remaining conjugates have modulus

We have strictly less than <1 Let’s wi for i=k+1,…n witch | wi |<1 the authors  AAroots of P. Show that Λ(Y) is irreducible. n−−1 ≤  nk |∑wik |=| | | w11 | <|ww |=| | ≤≤ AA  1 in nn By the condition of the Theorem, Λ(0)=λ0≠0 hence, all roots of the  AA |w w |=|n−−2 |≤ | ww | <|w w |=|nk | polynomial Λ(Y). are not equal to 0  ∑ ii12 12 1 k ≤≤ AA  1

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Then we have h−1 nh (−− 1) 1 |k (nh , ) |≤− ( q 1)( q − 1) q .

λn−1= AB sm −− 11+  Proof. The set of the minimal polynomials of the elements of λnk−+1= +++Ask−′′ +1 B mk − " AB sk − mk −"1 + k  (n,h) is λnk−= ++ABsk−−′ mk"  (nh , )={ PY ( )= Yn+ A Y n−−1 ++ A Ynk ++ A: This gives us that k n−−10nk

h= deg Ank− > supdegink≠−Ai anddeg Ank− = deg Ank−+11+ deg An− .} degλ − = sup(degA−− ;deg B ) = deg A − because| A − |=| w |>| w | for i ≠ 1  n1 sm1 1 s 1 s11 i degλnk− = degABsk −−′ + deg mk" By the consequence 3.5, we have hA=|nk− |>| A nk −+11 |> >| An− |>supink< − | Ai |.  λ ++ degnk−+1 = sup(degABsk−′′ degmk−"1 + ; deg Ask− +1 deg Bmk−" ). we obtain  h If degλnk−+1 =degABsk−′ + degmk−"1 + , then degAnk−− = h , then #Ank = ( q− 1) q  h−1 degAnk−+11≤− h 1, then # An− ≤q degλ =degAB′ + deg . n−−kskm −k"   And deg Ahkthen≤−+1, # Aq≤hk−+1  nn−−11 deg A≤− h k, then # A≤ qhk− degλλnk− =degnk−+11++ deg λn− degAn− 1 .  nk−−11n−   degABsk−−−−′′+ degmk" =deg ABsk ++ degmk"1+− deg As1 . hk− degA0≤−− h k 1, and A 00 ≠ 0 , then # A≤ q − 1  Also  Therefore degBBmk−" =degmk−+"1+ deg As−1 .

|k (nh , )| |BBmk−" |=| mk −+ "1 ||deg+ As−1 |. h h−1 hk −+1 ( hk − )( nk −+ 1) hk − This gives us that ≤−(q 1) qq . . q q( q − 1) h−1 ( hk − )( nk −+ 1) hk − [h+ ( h − 1) + ( hk −+ 1)]  ≤−(qq 1)( − 1) q ( q− 1) q |wwwwkkkmk′′+1 "|=| +1−+ "1|| w 1 |. k (2hk−+ 1) Therefore ≤−(qq 1)(h−1 − 1) q ( hk − )( nk −+ 1) ( q hk − − 1) q2 . |ww "|=| |. k 1 Conclusion With is absurd. The presentation and estimation for the number of the Pisot-k power formal series, precisely we will give their degrees and their Now if degλn−+ks1 =degAB−km′ + 1"+ deg−k , then logarithmic heights. degλn−−ks =degABkm′ + deg −k" . Acknowledgements And 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. degλλnk− =degnk−+11++ deg λn− degAn− 1 . References degABsk−′′+ degmk−" =deg Ask−+ 1 ++ deg Bmk− "1 deg As− . 1. Brauer A (1951) On algebraic equations with all but one root in the interior of This gives us that the unit circle. Math Nachr 4: 250-257.

|AAsk−′′ |=| sk −+11 |.|deg As− |. 2. Akiyama S (2006) Positive finiteness of number systems. Number theory , Tradition and Modernization. |AA |=| |.|deg A |. sk−′′ sk −+11s− 3. Bateman P, Duquette AL (1962) The analogue of Pisot-Vijayaraghavan We obtain numbers in fields of power series. Ill J Math 6: 594-406. 4. Chandoul A, Jellali M, Mkaouar M (2011) Irreducibility criterion over finite fields. |ww1kk′′ |=| ww 1− 11 || w |. Communication in Algebra 39: 3133-3137.

-1 Therefore 5. Chandoul A, Jellali, Mkaouar M (2013) The smallist Pisot element over q((X )), Communication in Algebra 56: 258-264. |ww |=| |. k′ 1 6. Sprindzuk VG (1963) Mahler’s problem in metric number theory, Translaion of Mathematical monographs. Amer math Soc. With is absurd. w1,...wk’+1…wk”

Let Λ(Y)=where Λi(Y),i=1,2,3 has of the coefficients in q [X].

Suppose in the first w1,...wk are the roots of Λ1 wk’+1…wk” are the roots of

Λ2, such that k′+k′′=k and the other roots are of Λ3. Clearly, the absolute value of leading coefficient of the polynomial3 Λ 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 k (n,h) the set of the element of Pisot-k series∈ in -1 q((X )) of degree n and of height logarithmic h Then the numbers of the elements of k (n,h) are

J Phys Math, an open access journal Volume 7 • Issue 3 • 1000194 ISSN: 2090-0902