Valérie Flammang

THE V-MEASURE FOR ALGEBRAIC INTEGERS HAVING ALL THEIR CONJUGATES IN A SECTOR

The v-measure for algebraic integers having all their conjugates in a sector

Introduction

Let α be a nonzero algebraic integer of degree d with conjugates α 1 = α, α 2 , . . . , α d . We define the V-measure of α by:

V(α) = d i=1 (1 + |α i |)(1 + 1/|α i |).
The absolute V-measure of α is the quantity v(α) = V(α) 1/d . If P is the minimal polynomial of α, then V(P ) = V(α) and v(P ) = v(α).

It is obvious that v(α) ≥ 4 and the equality holds if and only if α is a root of unity. Indeed, if for i = 1, . . . , d, we have (1 + |α i |)(1 + 1/|α i |) = 4 then for all i = 1, . . . , d, |α i | = 1. By the theorem of Kronecker, we deduce that α is a root of unity.

Remark: The scheme of this paper follows the scheme of our previous work [F2].

The case θ = 0

We suppose first that θ = 0. Then α is a totally positive algebraic integer i.e, all its conjugates are positive real numbers.

Theorem 1. For all nonzero totally positive algebraic integer α whose minimal polynomial P satisfies |P (0)| = 1, V(α) is a square. This is because, in this case, V(α) = P (-1) 2 . 1 Now, we want to study the structure of the set V of the quantities v(α) and prove the following Theorem 2. V is dense in (l, ∞) where l = lim n→∞ v(β 2 n ) = 5.649376 . . .. The β 2 n were defined by C.J. Smyth [START_REF] Smyth | The mean value of totally real algebraic numbers[END_REF] as follows:

β 2 0 = 1 β 2 n = β 2 n+1 + β -2 n+1 -2 β 2
n is a totally positive algebraic integer of degree 2 n . In order to determine the structure of the set V in the gap (4, l), we prove the following Theorem 3. If α is a nonzero totally positive algebraic integer whose minimal polynomial is different from x -1, x -2, x 2 -5x + 5, x 3 -7x 2 + 14x -7, x 4 -9x 3 + 27x 2 -31x + 11, x 4 -9x 3 + 26x 2 -28x + 9 and x 6 -13x 5 + 64x 4 -151x 3 + 177x 2 -96x + 19 then we have:

v(α) ≥ 4.935240.
Therefore, the seven smallest points of V in (4, l) are: 4 = v(x -1), 4.5 = v(x -2), 4.880511 = v(x 4 -9x 3 + 27x 2 -31x + 11), 4.919349 = v(x 2 -5x + 5), 4.925915 = v(x 6 -13x 5 + 64x

4 -151x 3 + 177x 2 -96x + 19, 4.932882 = v(x 4 -9x 3 + 26x 2 -28x + 9), 4.934380 = v(x 3 -7x 2 + 14x -7).
1.2 The case 0 < θ < 90 • Now, we consider algebraic integers α with all conjugates in a sector S θ = {z ∈ C such that | arg z| ≤ θ} with 0 < θ < 90 • . We follow the work of M. Langevin [La] on the absolute Mahler measure Ω(α) = ( d i=1 max(1, |α i |)) 1/d of algebraic integers α having all their conjugates in a sector. He proved that there exists a function c(θ) on [0, 180 • ), always > 1, such that if α = 0 is not a root of unity, whose conjugates all lie in S θ , then Ω(α) ≥ c(θ). His result is not numerical. G. Rhin and C. Smyth [RS] succeeded in finding the exact value of c(θ) for θ in nine subintervals of [0, 120 • ] and conjectured that c(θ) is a "staircase" decreasing function of θ, which is constant except for finitely many left discontinuities in any closed subinterval of [0, 180 • ). Their method uses explicit auxiliary polynomials A

(X) = X a R(X) where R is a reciprocal polynomial in Z Z[X],
found by heuristic methods. In 2004, thanks to Wu's algorithm [Wu], G. Rhin and Q. Wu [RW] gave the exact value of c(θ) for four new subintervals of [0, 140 • ] and extended four existing subintervals. In 2013, the author and G. Rhin [FR] found for the first time a complete subinterval and a fourteenth subinterval. A complete subinterval is an interval on which the function c(θ) describing the minimum on the sector | arg z| ≤ θ is constant, with jump discontinuities at each end. These improvements are due to our recursive algorithm.

We give in Table 3

a list of polynomials Q i with θ i = ϕ(Q i ) = max{| arg z| such that Q i (z) = 0}.
Now we define two functions f and g on [0, 90). The function g(θ) is the decreasing staircase function having left discontinuities at the angles θ i given in Table 3 and such that g(θ i ) = v(Q i ) . It gives the smallest known value of v(α) for α ∈ S θ then c(θ) ≤ g(θ). For 1 ≤ i ≤ 10, we define 10 non-increasing functions f i for θ ∈ [θ i , θ i+1 ] as follows:

f i (θ) = min z∈S θ   log(1 + |z|)(1 + 1/|z|) - 1≤j≤J c ij log |Q ij (z)|   .
The polynomials Q ij and the coefficients c ij can be read off from Table 3. The function

f is such that f (θ) = f i (θ) when θ ∈ [θ i , θ i+1 ) for 1 ≤ i ≤ 10. Since the functions f i are continuous we have f (θ) → f i (θ i+1 ) when θ → θ - i+1 .
We do not find any function f i such that f i (θ i ) > g(θ i ) for the other intervals [θ i , θ i+1 ), by Kronecker's theorem we may define f (θ) = 4 for θ ≥ θ 11 . Then the function f is non-increasing on [0, 90).

Theorem 4. The non-increasing functions f, g, c satisfy the following inequalities:

min(f (θ), g(θ)) ≤ c(θ) ≤ g(θ) (0 ≤ θ < 90).
Moreover, the exact value of c(θ) is known on ten subintervals of [0, 90).

These intervals are given in Table 1. One can read off the six intervals [θ i , θ i ) for 1 ≤ i ≤ 6, the interval [θ 8 , θ 8 ) and the three intervals [θ 7 , θ 7 ], [θ 9 , θ 9 ] and [θ 10 , θ 10 ] where c(θ) is known exactly. For θ in each of these intervals we have f (θ) > g(θ) so c(θ) = c(θ i ). Outside these intervals, c(θ) ≤ g(θ).

Table 1: The 10 intervals where c(θ) is known. The polynomials in the last column are the minimal polynomial of an algebraic integer belonging to S θ i and they are also listed in Table 3.

i c(θ) θ i θ i Q 1 4.
z 3 + z + 1 2 Denseness of the set V 2.1 Study of the sequence (v(β 2 n )) n≥1
As above the sequence (β 2 n ) n≥0 are defined as follows:

β 2 0 = 1 β 2 n = β 2 n+1 + β -2 n+1 -2
We first prove :

Lemma 5. v(β 2 n ) = 4 n-1 i=0 (1 + λ i ) 1/2 i where λ 0 = 1 4 and λ i+1 = λ i (1 + λ i ) 2 for i ≥ 0. Proof For n ≥ 0, we put γ n = β 2 n , so γ n = γ n+1 + γ -1 n+1 -2.
Therefore, we have:

V(β 2 n ) = V(γ n ) = 2 n i=1 (1 + γ n,i )(1 + γ -1 n,i ) = 2 n-1 i=1 ((1 + γ n,i )(1 + γ -1 n,i )) 2 because γ n,i = γ 2 n-1 +i for 1 ≤ i ≤ 2 n-1 . (For 1 ≤ i ≤ 2 n , γ n,i denotes the conjugates of γ n ). Thus: V(β 2 n ) = 2 n-1 i=1 (2 + γ n,i + γ -1 n,i ) 2 = 2 n-1 i=1 (4 + γ n-1,i ) 2 Finally, we have V(β 2 n ) = 4 2 n-1 2 n-1 i=1 (1 + 1/4γ n-1,i ) 2 .
We need here a more general lemma that we proved in [F1]:

Lemma 6. The notations are the same as previously.

2 n i=1 (1 + λ 0 γ n,i ) = n i=0 (1 + λ i ) 1/2 i 2 n where λ 0 = 1 4 and λ i+1 = λ i (1 + λ i ) 2 for i ≥ 0. Now, we can conclude that v(β 2 n ) = 4 n-1 i=0 (1 + λ i ) 1/2 i .
A short program with Pari (see Annex below) computes the limit l of the sequence v(β 2 n ) and we get l = 5.649376 . . .. The limit l also gives an upper bound for the first accumulation point of V.

Proof of Theorem 2

We follow the notations and proof of C.J. Smyth in [START_REF] Smyth | The mean value of totally real algebraic numbers[END_REF]. For a given function g

: [0, ∞) → R, put M(g) the set of all means M g (α) = 1 d d i=1 g(|α i |)
for α a totally real algebraic integer, i.e., all its conjugates

α 1 = α, • • • , α d are real numbers. When the limits exist, put a(g) = lim n→∞ M g (β n ) and c(g) = lim n→∞ M g (2 cos(2π/n)).
Our choice here for g is

g : x → log(1 + x 2 )(1 + 1/x 2 ) (2.1) because then M g (α) = log v(α 2 ).
The proof consists in two parts.

2.2.1 First step of the proof C.J. Smyth [START_REF] Smyth | The mean value of totally real algebraic numbers[END_REF] proved the following Theorem 7. Let g : R + → R + be a monotonic increasing function, zero on [0, 1]such that

lim x→∞ g(x + 1)/g(x) = 1
and the values of log 2 g(2k + 1) mod 1 (k=0,1,2,. . .) are everywhere dense in (0, 1).

Then the limit a(g) exists and M(g) is dense in (a(g), ∞).

We replace the function g defined by (3.1) by the function g * which satisfies the hypothesis of Theorem 7:

g * (x) =    g(x) if x > 1 0 if 0 ≤ x ≤ 1 (2.2) Since M g (β n ) = M g * (β n )
, the existence of a(g * ) implies those of a(g) and a(g * ) = a(g).

It is easy to see that g * satisfies the first hypothesis of Theorem 7. So, it is sufficient to study the denseness of the set F = {log 2 g(2k + 1) mod 1,k ∈ N}.

Let t ∈ [0, 1] and > 0. Does there exist f ∈ F such that |f (k) -t| < ? We search for n and k satisfying:

| log 2 g * (2k + 1) -t -n| < i.e., | log g * (2k + 1) -t -n log 2| < (2.3)
The uniform continuity of the function log on [1, ∞) gives :

∀ > 0, ∃ η( ) such that ∀x, y > 0, |x -y| < η( ) ⇒ | log x -log y| < .
We choose n such that 2 -n < η( ) and k such that

|(2k + 1) -(g * ) -1 (2 n e t )| ≤ 1. As (g * ) is
bounded by 1, the mean value Theorem for g * on (1, ∞) gives:

|g * (2k + 1) -2 n e t | ≤ 1, i.e., |2 -n g * (2k + 1) -e t | ≤ 2 -n < η( )
and the inequality (3.3) follows immediately. Thus, we have proved that For our function g defined by (3.1), the Lipschitz condition is satisfied for B(λ) = 2λ. Thus, M(g) is dense on (c(g), ∞) = (1.924847 . . . , ∞).

M(g) is dense in (a(g * ), ∞) = (a(g), ∞) = (log l, ∞) = (1.731545 . . . , ∞).

Conclusion

We have proved that M(g) is dense on (min(a(g), c(g)), ∞) = (log l, ∞) which means that V is dense on (l, ∞), where l = lim

n→∞ v(β 2 n ) = 5.649376 . . ..
3 Proof of Theorem 3

The explicit auxiliary function

The auxiliary function involved in Theorem 3 is of the following type:

for x > 0, f (x) = log(1 + x)(1 + 1 x ) - 0≤j≤J c j log |Q j (x)|
where the c j are positive real numbers and the polynomials Q j are non zero polynomials in

Z Z[x].
If α is a totally positive algebraic integer with conjugates α 1 = α,. . .,α d , we have

d i=1 f (α i ) ≥ md,
where m denotes the minimum of the function f . Thus,

log V(α) ≥ md + 1≤j≤J c j log | d i=1 Q j (α i )|.
Now assume that P does not divide any Q j , then

d i=1 Q j (α i ) is a nonzero integer because it is the resultant of P and Q j .
Therefore, if α is not a root of Q j , we have

v(α) ≥ e m .
The problem becomes to find a convenient set of polynomials Q j which gives a value of m as large as possible. This leads us to link the auxiliary function with the integer transfinite diameter in order to find the polynomials with our recursive algorithm.

Auxiliary functions and integer transfinite diameter

Let K be a compact subset of C.

If ϕ is a positive function defined on K, the ϕ-integer transfinite diameter of K is defined as

t Z,ϕ (K) = lim inf inf sup |P (x)| 1 n ϕ(x) . n ≥ 1 P ∈ Z[X] x ∈ K n → ∞ deg(P ) = n
This weighted version of the integer transfinite diameter was introduced by F. Amoroso [A].

Inside the auxiliary function (1), we replace the numbers c j by rational numbers a j /q where q is an integer such that q.c j is an integer for all 0 ≤ j ≤ J. Then we can write:

for x > 0, f (x) = log(1 + x)(1 + 1 x ) - t r log |Q(x)| (2)
where

Q = J i=0 Q a j j ∈ Z Z[X] is of degree r = J i=0 a j deg(Q j ) and t = J j=0 c j deg(Q j ) is a positive
real number. We want to get a function whose minimum m is as large as possible. It means to

search a polynomial Q ∈ Z Z[X] such that sup x>0 |Q(x)| t/r ((1 + x)(1 + 1/x)) -1 ≤ e -m .
If we suppose that t is fixed, it is clear that we need an effective upper bound for the quantity

t Z,ϕ ((0, ∞)) = lim inf inf sup |P (x)| t r ϕ(x) r ≥ 1 P ∈ Z[X] x > 0 r → +∞ deg(P ) = r
where we use the weight ϕ(x) = ((1 + x)(1 + 1/x)) -1 .

We replace the compact subset K by the infinite interval (0, ∞), but the weight ϕ ensures that the quantity t Z,ϕ ((0, ∞)) is finite.

Construction of the auxiliary function

We proceed as follows: suppose that we know Q 1 , Q 2 , ..., Q J . We use the semi-infinite linear programming (introduced in number theory by C. J. Smyth [START_REF] Smyth | On the measure of totally real algebraic numbers[END_REF]) to optimize f for this set of polynomials (i.e., to get the greatest possible m). We obtain the numbers c 1 , c 2 , ..., c J and f in the form (2) with t =

J i=1 c j deg(Q j ).
Then for several value of k, we seek a polynomial R(

x) = k l=0 a l x l ∈ Z Z[x] such that sup x>0 |Q(x)R(x)| t r+k ((1 + x)(1 + 1/x)) -1 ≤ e -m , i.e., such that sup x>0 |Q(x)R(x)|((1 + x)(1 + 1/x)) -(r+k)/t
is as small as possible.

We apply the LLL algorithm to the linear forms in a 0 ,. . ., a k

Q(x i )R(x i )((1 + x i )(1 + 1/x i )) -(r+k)/t .
The x i are control points uniformly distributed in the interval [0,70], including the points where f has its least local minima. We get a polynomial R whose factors R j are good candidates to extend the list of polynomials (Q 1 , Q 2 , ...,Q J ). We only keep the polynomials R j which have a nonzero coefficient c j in the new optimized auxiliary function f . After optimization, some previous polynomials Q j may have a zero coefficient and are removed.

In order to get the constant of Theorem 3, we take k from 4 to 20 successively.

The polynomials and the coefficients c j involved in the auxiliary function of Theorem 3 are listed in the Table 2 below.

Remark:

The main change with Wu's algorithm is that our polynomials are obtained by induction.

Proof of Theorem 4

Now, α is a nonzero algebraic integer with conjugates lie in a sector S θ = {z ∈ C such that | arg z| ≤ θ} with 0 < θ ≤ 90 • . The auxiliary functions f i , 1 ≤ i ≤ 10 are of the following type:

∀z ∈ S θ , f (z) = log(1 + |z|)(1 + 1/|z|) - 1≤j≤J c j log |Q j (z)|,
where the c j are positive real numbers and the polynomials Q j are non zero polynomials in Z Z[z].

Since the function f is harmonic outside a finite set containing the roots of the polynomials Q j , it takes its minimum on the boundary of S θ . It is clear that f (z) → ∞ as z → ∞. Thus, as we have f (z) = f (z), it is sufficient to search the minimum of f on the upper edge of the sector S θ where z = xe iθ with x > 0.

The auxiliary function on the half line

R θ = {z ∈ C, z = xe iθ , x > 0} is f (z) = log(1 + x)(1 + 1 x ) - 1≤j≤J c j log |Q j (z))|.
Concerning the integer transfinite diameter, we replace here the compact K by the half line R θ . But, again, the weight ϕ(z) = (|z| + 1/|z|) -1 ensures that the quantity t Z Z,ϕ (R θ ) is a finite number.

Then, for several values of k, we search a polynomial

R(z) = k l=0 a l z l ∈ Z Z[z] such that sup x>0 |Q(z)R(z)| t r+k |(1 + x)(1 + 1/x)| -1 ≤ e -m ,
where m is the minimum of the function f . We search

sup x>0 |Q(z)R(z)||(1 + x)(1 + 1/x)| -(r+k)/t
is as small as possible.

Since R(z) is no more a real linear form in the unknown coefficients a i , we replace it by its real part and its imaginary part. Then, we apply LLL to the linear forms

|Q(z n ).Re(R(z n ))•|(1+x n )(1+1/x n )| -(r+k)/t and |Q(z n )•Im(R(z n )).|(1+x n )(1+1/x n )| -(r+k)/t .
The x n are suitable control points in [0, 50] , including the points wheref has its least local minima. Then we apply our recursive algorithm as described in Section 3.

The polynomials and the coefficients c j involved in the auxiliary function of Theorem 4 are listed in the Table 3 below.

Annex: The program which computes lim

n→∞ v(β 2 n ) N=25; lambda=vector(N,j,0); lambda[1]=1/4; for(k=2,N,lambda[k]=lambda[k-1]/(1+lambda[k-1]) 2 );p=4;for(k=1,N-1,p=p*(1+lambda[k]) 1/2 k-1 ;print(p*1.))
The computation gives:

j c j Q j 1 0.134139923394118 x -1 2 0.061749635022796 x -2 3 0.009744089471627 x -3 4 0.001848231653815 x 2 -3 + 1 5 0.008718157080652 x 2 -4 + 2 6
0.004623942020398 x 2 -5 + 5 7 0.001677775683442 x 3 -6x 2 + 9 -3 8 0.000770087575792 x 3 -7x 2 + 14 -7 9 0.003025585967283 x 4 -9x 3 + 26x 2 -28 + 9 10 0.008560749688132

x 4 -9x 3 + 27x 2 -31 + 11 11 0.004698394045964

x 6 -13x 5 + 64x 4 -151x 3 + 177x 2 -96 + 19 12 0.001126497871938

x 6 -13x 5 + 65x 4 -158x 3 + 193x 2 -110 + 23 13 0.000001874362705

x 6 -13x 5 + 65x 4 -158x 3 + 194x 2 -113 + 25 14 0.000403173239206

x 6 -14x 5 + 76x 4 -203x 3 + 278x 2 -182 + 43 15 0.000383084810882

x 7 -16x 6 + 102x 5 -333x 4 + 596x 3 -580x 2 + 282 -53 16 0.000260704467608

x 8 -18x 7 + 134x 6 -536x 5 + 1252x 4 -1736x 3 + 1385x 2 -578 + 97 17 0.000745912646983

x 8 -18x 7 + 134x 6 -537x 5 + 1260x 4 -1760x 3 + 1418x 2 -598 + 101 18 0.000652672384412

x 8 -18x 7 + 135x 6 -547x 5 + 1299x 4 -1835x 3 + 1492x v(Q j ) c j θ(Q j ) Q j 4.00000 0.120028 0 x -1 4.30940 0.012943 30.0000

2 -3x + 3 f 5 v(Q j ) c j θ(Q j ) Q j 4.
x 2 -3x + 3 4.32814 0.001299 34.5817

x 6 -8x 5 + 29x 4 -58x 3 + 68x 2 -44x + 13 4.31264 0.004190 35.6858

x 8 -10x 7 + 47x 6 -131x 5 + 236x 4 -280x 3 + 215x 2 -98x + 21 4.27300 0.008190 36.2431

x 4 -5x 3 + 11x -11x + 5 4.35739 0.001903 37.7612

x 4 -5x 3 + 11x 2 -10x + 4 4.12132 0.019883 45.0000

x 2 -2x + 2 f 7 v(Q j ) c j θ(Q j ) Q j 4.00000 0.06036 0 x -1 4.27300 0.008216 36.2431

x 4 -5x 3 + 11x 2 -11x + 5 4.15075 0.003199 39.0700 2x 6 -11x 5 + 29x 4 -43x 3 + 39x 2 -20x + 5 4.34442 0.001774 39.4782

x 6 -7x 5 + 23x 4 -40x 3 + 41x 2 -23x + 6 4.36481 0.000167 41.8833

x 7 -9x 6 + 33x 5 -69x 4 + 88x v(Q j ) c j θ(Q j ) Q j 4.00000 0.036463 60.0000

x 2 -x + 1 4.11491 0.012541 61.1985

x 4 -x 3 + 2x 2 + 1 f 10 v(Q j ) c j θ(Q j ) Q j 4.07339 0.007302 73.6316

x 3 + x + 1 4.20906 0.008674 79.0744

x 8 -2x 7 + 10x 6 -12x 5 + 28x 4 -19x 3 + 26x 2 -8x + 7 4.21560 0.009987 80.2809

x 4 -x 3 + 5x 2 -2x + 4 4.20719 0.000880 83.1995

x 6 -x 5 + 7x 4 -4x 3 + 13x 2 -3x + 6 4.11803 0.013675 84.2506

x 7 -2x 6 + 7x 5 -10x 4 + 14x 3 -14x 2 + 8x -5 4.12132 0.010461 90.0000

x 2 + 2

  2.2.2 Second step of the proof C.J. Smyth [Sm1] established the following Theorem 8. Let g : R + →: R + be a function such that lim x→∞ g(x) = ∞ and which satisfies a Lipschitz condition |g(x) -g(y)| < B(λ)|x -y| for x, y ∈ [0, λ], for each λ > 0. Then M(g) is dense on (c(g), ∞), where ccos θ)dθ.

Table 2 :

 2 The polynomials involved in Theorem 3.

	5.0000000000000
	5.3851648071345
	5.5385668995107
	5.6017408320507
	5.6284946631685
	5.6400801272366
	5.6451863873055
	5.6474688546314
	5.6485007769183
	5.6489716924285
	5.6491882700948
	5.6492885311707
	5.6493352061849
	5.6493570404592
	5.6493672976162
	5.6493721340930
	5.6493744221264
	5.6493755077355
	5.6493760241907
	5.6493762704706
	5.6493763881680
	5.6493764445270
	5.6493764715632
	5.6493764845544

  2 -633 + 107 19 0.000352222775710 x 8 -19x 7 + 149x 6 -627x 5 + 1539x 4 -2239x 3 + 1868x 2 -809 + 139 20 0.000068815531942 x 8 -19x 7 + 149x 6 -627x 5 + 1541x 4 -2254x 3 + 1907x 2 -850 + 153 21 0.000014696986402 x 12 -28x 11 + 344x 10 -2451x 9 + 11268x 8 -35159x 7 + 76200x 6 -115315x 5 +120628x 4 -84845x 3 + 38001x 2 -9718 + 1073

Table 3 :

 3 The auxiliary functions f i , 1 ≤ i ≤ 10 -7x 3 + 19x 2 -22x + 10 4.44065 6.45845 E-5 26.2404 x 12 -19x 11 + 168x 10 -910x 9 + 3354x 8 -8846x 7 + 17102x 6 -24411x 5

	f 1	v(Q j )	c j	θ(Q j )	Q j
		4.50000 0.109569	0	x -2
		4.00000 0.159943	0	x -1
		4.49390 0.001016	20.5575	x 8 -13x 7 + 74x 6 -239x 5 + 476x 4 -595x 3 + 454x 2 -194x + 37
		4.47076 0.001195	22.3862	x 4 -7x 3 + 19x 2 -23x + 11
		4.53330 0.000250	22.7179	x 10 -17x 9 + 131x 8 -7 + 1793x 6 -3659x 5 + 5140x 4 -4894x 3
					+3016x 2 -1086x + 175
		4.58284 9.89994 E-5 23.1681 x 5 -9x 4 + 33x 3 -60x 2 + 53x -17
		4.64998 0.002097	23.7496	x 4 -8x 3 + 25x 2 -35x + 19
		4.53248 0.000469	24.0255	x 10 -17x 9 + 131x 8 -599x 7 + 1792x 6 -3651x 5 + 5112x 4 -4840x 3
					+2956x 2 -1050x + 166
		4.52195 0.000883	24.4961	x 10 -17x 9 + 131x 8 -599x 7 + 1793x 6 -3660x 5 + 5146x 4 -4909x 3
					+3035x 2 -1098x + 178
	f 2	v(Q j )	c j	θ(Q j )	Q j
		4.00000 0.140130	0	x -1
		4.49390 0.004873	20.5575	x 8 -13x 7 + 74x 6 -239x 5 + 476x 4 -595x 3 + 454x 2 -194x + 37
		4.47076 0.008004	22.3862	x 4 -7x 3 + 19x 2 -23x + 11
		4.52195 0.000301	24.4961	x 1 0 -17x 9 + 131x 8 -599x 7 + 1793x 6 -3660x 5 + 5146x 4 -4909x 3
					+3035x 2 -1098x + 178
		4.30940 0.029232	30.0000	x 2 -3x + 3
	f 3	v(Q j )	c j	θ(Q j )	Q j
		4.00000 0.156836	0	x -1
		4.30940 0.023194	30.0000	x 2 -3x + 3
		4.47076 0.004332	22.3862	x 4 -7x 3 + 19x 2 -23x + 11
		4.53161 5.86076 E-5 25.5367 x 4 +25540x 4 -19123x 3 + 9750x 2 -3053x + 448
		4.48215 0.000362	26.4803	x 12 -20x 11 + 186x 10 -1059x 9 + 4100x 8 -11350x 7 + 23008x 6 -34386x 5
					+37591x 4 -29318x 3 + 15494x 2 -4989x + 743
		4.68328 0.006789	26.5651	x 2 -4x + 5
		4.44726 0.000805	27.4317	x 8 -13x 7 + 76x 6 -258x 5 + 554x 4 -769x 3 + 675x 2 -344x + 79
		4.47572 0.000172	27.5096	x 12 -20x 11 + 186x 10 -1059x 9 + 4099x 8 -11338x 7 + 22944x 6 -34191x 5
					+37224x 4 -28887x 3 + 15191x 2 -4876x + 727
		4.45763 0.000147	28.2826	x 12 -19x 11 + 168x 10 -909x 9 + 3340x 8 -8758x 7 + 16778x 6 -23643x 5
					+24325x 4 -17837x 3 + 8867x 2 -2693x + 381
		4.44764 0.001217	28.3244	x 6 -10x 5 + 43x 4 -100x 3 + 132x 2 -94x + 29
		4.46165 0.000199	28.3538	x 12 -19x 11 + 168x 10 -909x 9 + 3340x 8 -8757x 7 + 16767x 6 -23592x 5
					+24196x 4 -17646x 3 + 8703x 2 -2619x + 368
		4.48237 0.000966	28.9238	x 7 -12x 6 + 64x 5 -194x 4 + 359x 3 -404x 2 + 256x -71
		4.49944 0.000320	29.9530	x 13 -22x 12 + 225x 11 -1414x 10 + 6082x 9 -18880x 8 + 43463x 7 -75057x 6
					+97177x 5 -93148x 4 + 64276x 3 -30269x 2 + 8738x -1173
		4.50347 0.000233	30.9258	x 10 -17x 9 + 133x 8 -626x 7 + 1954x 6 -4212x 5 + 6334x 4 -6550x 3
					+4454x 2 -1799x + 329
	f 4	v(Q j )	c j	θ(Q j )	Q j
		4.00000 0.160737	0	x -1
		4.47076 0.004736	22.3862	x 4 -7x 3 + 19x 2 -23x + 11
		4.68328 0.007227	26.5651	x 2 -4x + 5
		4.44726 0.001341	27.4317	x 8 -13x 7 + 76x 6 -258x 5 + 554x 4 -769x 3 + 675x 2 -344x + 79
		4.43050 0.000275	27.8282	x 12 -19x 11 + 168x 10 -910x 9 + 3354x 8 -8847x 7 + 17112x 6 -24455x 5
					+25648x 4 -19281x 3 + 9886x 2 -3116x + 460
		4.44764 0.001702	28.3244	x 6 -10x 5 + 43x 4 -100x 3 + 132x 2 -94x + 29
		4.46165 0.000953	28.3538	x 12 -19x 11 + 168x 10 -909x 9 + 3340x 8 -8757x 7 + 16767x 6 -23592x 5
					+24196x 4 -17646x 3 + 8703x 2 -2619x + 368
		4.48237 0.001132	28.9238	x 7 -12x 6 + 64x 5 -194x 4 + 359x 3 -404x 2 + 256x -71
		4.30940 0.020855	30.0000	x

  -12x 7 + 65x 6 -205x 5 + 410x 4 -531x 3 + 436x 2 -209x + 46 4.40730 0.001488 34.0856 x 8 -12x 7 + 66x 6 -213x 5 + 438x 4 -585x 3 + 496x 2 -245x + 55 f 6

	00000 0.149253	0	x -1
	4.44726 0.000224	27.4317	x 8 -13x 7 + 76x 6 -258x 5 + 554x 4 -769x 3 + 675x 2 -344x + 79
	4.43050 0.000276	27.8282	x 12 -19x 11 + 168x 10 -910x 9 + 3354x 8 -8847x 7 + 17112x 6 -24455x 5
			+25648x 4 -19281x 3 + 9886x 2 -3116x + 460
	4.44764 0.000758	28.3244	x 6 -10x 5 + 43x 4 -100x 3 + 132x 2 -94x + 29
	4.30940 0.024490	30.0000	x 2 -3x + 3
	4.37789 0.007228	32.0950	x 4 -6x 3 + 15x 2 -17x + 8
	4.39727 0.001220	34.0406	x 8

  3 -70x 2 + 32x -7 4.12132 0.02604 45.0000 x 2 -2x + 2 4.21656 0.01509 45.3100 x 4 -4x 3 + 8x 2 -7x + 3 4.20509 0.000723 63.2729 x 6 -5x 5 + 15x 4 -26x 3 + 28x 2 -17x + 5

	f 8	v(Q j )	c j	θ(Q j )	Q j
		4.12132 0.05118	45.0000 x 2 -2x + 2
		4.00000 0.03096	60.0000 x 2 -x + 1
	f 9


V Flammang

COMPARISON OF THE MAHLER MEASURE AND THE LENGTH FOR ALGEBRAIC INTEGERS HAVING ALL THEIR CONJUGATES IN A SECTOR

In this paper, explicit auxiliary functions are used to get lower and upper bounds for the Mahler measure involving the length for an algebraic integer having all its conjugates in a sector. Mostly, these bounds improve the usual ones. At our knowledge, it is the first time that such a work is done.

Introduction

Let α be a nonzero algebraic number of degree d with minimal polynomial P = a 0 x d + a 1 x d-1 + ... + a d = a 0 (x -α 1 )...(x -α d ). We define:

• the Mahler measure of α as M(α) = We have the well-known inequalities : 2 -d L(α) ≤ M(α) ≤ L(α). Now, we consider that α is an algebraic integer whose all conjugates lie in a sector S θ = {z ∈ C such that | arg z| ≤ θ}, with 0 ≤ θ ≤ 90 • . Since the algebraic integers α 1 = α, ..., α d have positive real parts, we can show by induction that then the polynomial P has a sequence of coefficients with alternate signs so L(α) = |P (-1)| =

d i=1 |1 + α i |.
In 2014, we studied the case of θ = 0 [F2]. It means that the conjugates of α are all positive real numbers. Using the method of explicit auxiliairy functions and our recursive algorithm, we proved:

Theorem 1. If the minimal polynomial P of α is different from x, x -1, x 2 -3x + 1, x 4 -7x 3 + 13x 2 -7x + 1, x 2 -4x + 1, x 6 -12x 5 + 44x 4 -67x 3 + 44x 2 -12x + 1 and x 8 -15x 7 + 83x 6 -220x 5 + 303x 4 -220x 3 + 83x 2 -15x + 1 then 1.058358 d L(α) 0.562454 ≤ M(α) ≤ 0.379128 d L(α) 1.803995 . These inequalities are generally better than the classical ones. For instance, if α has minimal polynomial x 3 -6x 2 + 8x -1, we get:

1.058358 d L(α) 0.562454 = 5.63845 ≤ M(α) = 7.18421 ≤ 8.10185 = 0.379128 d L(α) 1.803995 1 instead of 2 -d L(α) = 2 ≤ M(α) = 7.18421 ≤ L(α) = 16.
The purpose of this paper is to establish such inequalities in the case of 0 < θ ≤ 90. It is the first time that this kind of work is done at our knowledge. We need to recall a theoretical result of M. Langevin [La] on the Mahler measure. He proved that there exists a function c(θ) on [0, 180 • ), always > 1, such that if α = 0 is not a root of unity, whose conjugates all lie in S θ , then M(α) 1/d ≥ c(θ) where d denotes the degree of α. In 1995, G. Rhin and C. Smyth [RS] were the first to succeed in finding the exact value of c(θ) for θ in nine subintervals of [0, 120 • ] and conjectured that c(θ) is a "staircase" decreasing function of θ, which is constant except for finitely many left discontinuities in any closed subinterval of [0, 180 • ). They used the method of explicit auxiliary functions with polynomials found by heuristic search. Later, in 2004, G. Rhin and Q. Wu [RW] gave the exact value of c(θ) for four new subintervals of [0, 140 • ] and extended four existing subintervals. These improvements were allowed thanks to Wu's algorithm [Wu]. Based on this algorithm we have developed our recursive algorithm where the polynomials are found by induction. It allowed the author and G. Rhin [FR] in 2013 to find for the first time a complete subinterval and a fourteenth subinterval. A complete subinterval is an interval on which the function c(θ) describing the minimum on the sector | arg z| ≤ θ is constant, with jump discontinuities at each end. In 2014 [F3], we applied this reasoning to the length. It means that we computed the greatest lower bound c(θ) of the quantities L(α) 1/d for θ belonging to eight subintervals of [0,90). Here again, one subinterval is complete. The following table summarizes our results: From now on, we suppose that the conjecture of G. Rhin and C. Smyth on the function c(θ) is true. We prove the following results:

Theorem 2. If α is an algebraic integer whose conjugates all lie in: We give in Table 2 below an example for each sector.

[0, 18.26) then 1.141148 d .L(α) 0.427453 ≤ M(α) ≤ 0.218243 d .L(α) 2.490525 [18.26, 29.78) then 0.681486 d .L(α) 0.814756 ≤ M(α) ≤ 0.343833 d .L(α)
The proof of these different inequalities uses the principle of explicit auxiliary function introduced into Number Theory by C. J. Smyth [START_REF] Smyth | The mean value of totally real algebraic numbers[END_REF]. The method is based on the fact that the resultant of two polynomials in Z Z[X] with no common roots is a nonzero integer.

To get the lower bounds, the auxiliary function is of the following type:

∀z ∈ S θ , f (z) = log max(1, |z|) -c 0 log |z + 1| - 1≤j≤J c j log |Q j (z)| (1.1)
where the c j s are positive real numbers and the polynomials

Q j are nonzero elements of Z Z[X]. Then d i=1 f (α i ) ≥ md,
where m denotes the minimum of the function f on the sector S θ . Thus

log M(α) ≥ md + c 0 log L(α) + 1≤j≤J c j log | d i=1 Q j (α i )|.
If the minimal polynomial P of α does not divide any

Q j then d i=1 Q j (α i ) is a nonzero integer
because it is the resultant of P and Q j . In this case, we have

M(α) ≥ e md L(α) c 0 .
To get the upper bounds, the auxiliary function is of the following type:

∀z ∈ S θ , f (z) = -log max(1, |z|) + c 0 log |z + 1| - 1≤j≤J c j log |Q j (z)|.
In each case, f is invariant under complex conjugation. We can then limit ourselves to 0 ≤ arg z ≤ θ. Since the function f is harmonic outside the union of arbitrary small disks around the roots of the polynomials Q j , the minimum is taken on the upper edge of S θ where z = xe iθ with x > 0.

The auxiliary function on the half line

R θ = {z ∈ C, z = xe iθ , x > 0} is f (z) = log max(1, x) -c 0 log |z + 1| - 1≤j≤J c j log |Q j (z)|. (1.2)
In the following sections, we explain how to construct an auxiliary function, especially how to find a good set of polynomials Q j . All the computations are done on a MacBook Pro with the langages Pascal and Pari.

2 Construction of an explicit auxiliary function

Rewriting the auxiliary function (1.2)

Inside this auxiliary function, we replace the numbers c j by rational numbers a j /q where q is an integer such that q.c j is an integer for all 1 ≤ j ≤ J. Then we have

∀x > 0, f (z) = log max(1, x) -c 0 log |z + 1| - t r log |Q(z)| ≥ m (2.1)
where

Q = J i=0 Q a j j ∈ Z Z[X] is of degree r = J i=1 a j deg(Q j ) and t = J j=1 c j deg(Q j ) is a positive
real number. We want to get a function whose minimum m is as large as possible. It means to

search a polynomial Q ∈ Z Z[X] such that sup x>0 |Q(z)| t/r |z + 1| c 0 max(1, x) ≤ e -m .
If we suppose that t is fixed, we need to get an effective upper bound for the quantity

t Z,ϕ ((0, ∞)) = lim inf inf sup |P (z)| t r ϕ(z) r ≥ 1 P ∈ Z[X] x > 0 r → +∞ deg(P ) = r .
where we use the weight ϕ

(z) = |z + 1| c 0 max(1, x) .
It is clear that this quantity is closely related to the weighted version of the integer transfinite diameter of a compact subset

K of C t Z,ϕ (K) = lim inf inf sup |P (z)| 1 n ϕ(z) . n ≥ 1 P ∈ Z[z] z ∈ K n → ∞ deg(P ) = n
where ϕ is a positive function defined on K. We replace K by the infinite interval (0, ∞), but the weight ϕ ensures that the quantity t Z,ϕ ((0, ∞)) is finite.

How to find the polynomials Q j

We proceed as follows: suppose that we know Q 1 , Q 2 , ..., Q J . We use the semi-infinite linear programming method (introduced into number theory by C. J. Smyth [START_REF] Smyth | On the measure of totally real algebraic numbers[END_REF]) to optimize f for this set of polynomials (i.e., to get the greatest possible m). We obtain the numbers c 0 , c 1 , c 2 , ..., c J and f in the form (1.2) with t = J i=1 c j deg(Q j ).

Then, for several values of k, we search a polynomial R

(z) = k l=0 a l z l ∈ Z Z[z] such that sup x>0 |Q(z)R(z)| t r+k |z + 1| c 0 max(1, x) ≤ e -m ,
where m is the minimum of the function f . We search

sup x>0 |Q(z)R(z)| |z + 1| c 0 max(1, x) -(r+k)/t
is as small as possible.

Since R(z) is not a real linear form in the unknown coefficients a i , we replace it by its real part and its imaginary part. Then, we apply LLL to the linear forms

|Q(z n ).Re(R(z n )) • |z n + 1| c 0 max(1, x n ) -(r+k)/t and |Q(z n ) • Im(R(z n )). |z n + 1| c 0 max(1, x n ) -(r+k)/t
. The x n are suitable control points in [0, 50] , including the points where f has its least local minima. Remember that z n = x n e iθ . We get a polynomial R whose factors R j are good candidates to extend the list of polynomials (Q 1 , Q 2 , ...,Q J ). We only keep the polynomials R j which have a nonzero coefficient c j in the new optimized auxiliary function f . After optimization, some previous polynomials Q j may have a zero coefficient and are removed.

The polynomials Q j and their coefficients c j can be read off from Table 3 and Table 4 below.

Where the Rhin-Smyth conjecture is needed

Within our semi-infinite linear programming program, we need a lower bound for the quantities L(α) 1/d where α is an algebraic integer having all its conjugates in a sector. Remember that the function c(θ) is exactly the greatest lower bound of these quantities. That is why the eight subintervals of [0, 90) appear in Theorem 2. Because on each of them, either we know the exact value of c(θ), or, under the hypothesis that the conjecture of G. Rhin and C. Smyth is true, we have a lower bound for it thus for the quantities L(α) 1/d . We give an example of how it works.

For θ ∈ [0, 18.26), we have [F3] c(θ) = 2.236068. But we do not know the exact value of c(θ) on [18.26, 26.408740). Nevertheless, if we suppose that c(θ) is a staircase decreasing function of θ, then we can say that c(θ) ≥ 2.140695 which is the exact value of c(θ) on the following subinterval [26.408740, 29.78). We iterate this reasoning on each subinterval where the exact value of c(θ) is not known. x 3 -x 2 + 2x -1 0.625000 1.54279 1.75488 3.14369 5.00000

Table 3: The polynomials Q j and their coefficients c j used for the lower bounds in Theorem 2.

[0, 18.26)

c j Q j 0.069302390379 x 0.371118419047 x -1 0.031411836366 x 2 -3x + 1 [18.26, 29.78) c j Q j 0.002437560806 5x 4 -18x 3 + 29x 2 -23x + 8 0.001061948615 5x 4 -19x 3 + 32x 2 -26x + 9 0.009415541713 10x 4 -34x 3 + 49x 2 -34x + 10 0.000307427745 13x 4 -45x 3 + 65x 2 -45x + 13 0.008285897341 2x 7 + 9x 6 -81x 5 + 224x 4 -322x 3 + 265x 2 -120x + 24 0.004756616202 x 9 + 71x 8 -509x 7 + 1633x 6 -3106x 5 + 3835x 4 -3148x 3 + 1681x 2 -536x +79 0.000727997673 2x 9 -111x 8 + 741x 7 -2383x 6 + 4623x 5 -5858x 4 + 4948x 3 -2721x 2 +893x -135 0.001049319352 x 11 -200x 10 + 1715x 9 -7034x 8 + 17843x 7 -30825x 6 + 37797x 5 -33283x 4 +20788x 3 -8827x 2 + 2308x -284 0.001949219424 2x 11 -195x 10 + 1618x 9 -6542x 8 + 16471x 7 -28360x 6 + 34767x 5
-30686x 4 + 19251x 3 -8225x 2 + 2167x -269 0.001033849131 3x 11 -205x 10 + 1703x 9 -6988x 8 + 17893x 7 -31307x 6 + 38924x 5 -34753x 4 + 21991x 3 -9447x 2 + 2494x -309 0.000521463318 5x 11 -363x 10 + 3023x 9 -12303x 8 + 31050x 7 -53315x 6 + 64844x 5 -56497x 4 + 34818x 3 -14542x 2 + 3726x -447 0.003314331646 11x 13 -775x 12 + 7505x 11 -36392x 10 + 111965x 9 -240826x 8 + 379836x 7 -449505x 6 + 401780x 5 -269162x 4 + 131873x 3 -44910x 2 + 9567x -968 0.000216099044 12x 13 -817x 12 + 7920x 11 -38487x 10 + 118612x 9 -255348x 8 + 402718x 7 -476099x 6 + 424719x 5 -283724x 4 + 138504x 3 -46965x 2 + 9956x -1002 0.000861280624 13x 13 -797x 12 + 7747x 11 -38024x 10 + 118599x 9 -258563x 8 + 412985x 7 -494313x 6 + 446216x 5 -301407x 4 + 148633x 3 -50851x 2 + 10860x -1099 [29.78, 40.42) [40.42, 48.68) [60, 71.19) c j Q j 0.177813299876 x 2 + 1 0.022335665277 2x 2 -x + 2 0.019836674768 x 6 -x 5 + 5x 4 -3x 3 + 5x 2 -x + 1 0.005717360870 3x 6 -4x 5 + 12x 4 -9x 3 + 12x 2 -4x + 3 0.001223177207 x 10 -3x 9 + 18x 8 -33x 7 + 65x 6 -67x 5 + 70x 4 -40x 3 + 24x 2 -6x + 2 [71.19, 81.02) c j Q j 0.004602700691 x 6 -x 5 + 5x 4 -3x 3 + 5x 2 -x + 1 0.085737005206 3x 6 -4x 5 + 12x 4 -9x 3 + 12x 2 -4x + 3 Table 4: The polynomials Q j and their coefficients c j used for the upper bounds in Theorem 2.

c j Q j 0.215858078320 x 2 -x + 1 0.022246718914 2x 2 -3x + 2 0.001855528030 x 4 -3x 3 + 5x 2 -3x + 1 0.001442212353 x 4 -4x 3 + 8x 2 -7x + 3 0.001307301497 3x 4 -7x 3 + 8x 2 -4x + 1 0.000415058230 3x 4 -9x 3 + 13x 2 -9x + 3 0.001852500229 x 6 -5x 5 + 13x 4 -17x 3 + 13x 2 -5x + 1 0.000492561336 x 8 -9x 7 + 39x 6 -98x 5 + 159x 4 -171x 3 + 121x 2 -52x + 11 0.000120050649 x 8 -9x 7 + 39x 6 -100x 5 + 168x 4 -188x 3 + 138x 2 -61x + 13 0.000342003517 2x 8 -12x 7 + 38x 6 -72x 5 + 89x 4 -72x 3 + 38x 2 -12x + 2 0.001681915713 x 10 -3x 9 + 20x 7 -52x 6 + 69x 5 -52x 4 + 20x 3 -3x + 1 0.000125294152 x 10 -11x 9 + 60x 8 -200x 7 + 450x 6 -711x 5 + 801x 4 -637x 3 + 345x 2 -116x + 19 0.001910181120 5x 12 -50x 11 + 248x 10 -773x 9 + 1668x 8 -2607x 7 + 3019x 6 -2607x 5 +1668x 4 -773x 3 + 248x 2 -50x + 5
c j Q j 0.02590718645 x 2 -2x + 2 0.01941817601 2x 2 -2x + 1 0.11827937410 2x 2 -3x + 2 0.00038026400 4x 4 -9x 3 + 10x 2 -5x + 1 0.01060190670 7x 6 -29x 5 + 57x 4 -65x 3 + 46x 2 -19x + 4 [48.68, 54.93) c j Q j 0.155106991371 2x 2 -3x + 2 0.008668166571 2x 5 -8x 4 + 14x 3 -11x 2 + 4x + 1 0.003912861880 6x 7 -44x 6 + 127x 5 -215x 4 + 230x 3 -161x 2 + 68x -15 0.004034414973 9x 7 -35x 6 + 78x 5 -98x 4 + 82x 3 -40x 2 + 12x -1 0.004310120971 4x 8 -33x 7 + 124x 6 -273x 5 + 395x 4 -384x 3 + 252x 2 -102x + 22 0.003806593612 8x 8 -49x 7 + 148x 6 -269x 5 + 324x 4 -261x 3 + 139x 2 -44x + 7 0.001615690909 6x 9 -62x 8 + 235x 7 -539x 6 + 809x 5 -851x 4 + 620x 3 -311x 2 + 96x -16 [54.93, 60) c j Q j 0.00000010000 x 4 -2x 3 + 4x 2 -2x + 1 0.00000010000 x 4 -3x 3 + 6x 2 -4x + 2 0.00000010000 2x 4 -4x 3 + 6x 2 -3x + 1 0.00000010000 2x 4 -4x 3 + 7x 2 -4x + 2 0.00000010000 x 6 -x 5 + 4x 3 -5x 2 + 3x -1 0.00000010000 x 6 -9x 5 + 25x 4 -41x 3 + 37x 2 -19x + 5 0.00000010000 x 7 -29x 6 + 412x 5 -1091x 4 + 1844x 3 -1589x 2 + 813x -198 0.00000010000 x 8 -48x 7 + 2076x 6 -8817x 5 + 21913x 4 -30384x 3 + 27630x 2 -13250x +3929
[0, 18.26) c j Q j 0.647286246321 x 0.097976553424 x 2 -3x + 1 [18.26, 29.78) c j Q j 0.474121906237 x 0.010151342146 x 4 -6x 3 + 12x 2 -6x + 1 0.000222468351 2x 7 -23x 6 + 94x 5 -177x 4 + 144x 3 -59x 2 + 12x -1 [29.78, 40.42) c j Q j 0.678499578979 x [40.42, 48.68) c j Q j 1.13905764949 x 0.04916744121

x 4 -3x 3 + 5x 2 -3x + 1 0.01162652604

x 6 -5x 5 + 13x 4 -17x 3 + 13x 2 -5x + 1 [48.68, 54.93) c j Q j 0.878492951062 x 0.029955915957 x 4 -3x 3 + 5x 2 -3x + 1 0.017156080155 x 4 -3x 3 + 6x 2 -3x + 1 0.003933620090 x 4 -3x 3 + 7x 2 -3x + 1 0.006942288287 x 8 -6x 7 + 21x 6 -41x 5 + 53x 4 -41x 3 + 21x 2 -6x + 1 [54.93, 60) c j Q j 1.092450513855 x 0.045279700135 x 4 -2x 3 + 4x 2 -2x + 1 0.022451405283 x 8 -5x 7 + 16x 6 -29x 5 + 37x 4 -29x 3 + 16x

  |α i |), • the length of α as L(α) = d i=0 |a i |.

Table 1

 1 Intervals [θ i , θ i ] for i = 6 and [θ 6 , θ 6 ) where c(θ) is known exactly. Here c(θ) = c(θ i ) = L(P ) 1/ deg P for θ ∈ [θ i , θ i ] when i = 6 and for θ ∈ [θ 6 , θ 6 ).

	i	c(θ)	θ i	θ i	P
	1 2.236068	0	18.26 x 2 -3x + 1
	2 2.140695 26.408740 29.78	x 4 -5x 3 + 9x 2 -5x + 1
	3 2.030543 38.668282 40.42	x 4 -4x 3 + 7x 2 -4x + 1
	4 1.950116 47.941432 48.68	x 8 -7x 7 + 25x 6 -50x 5 + 63x 4 -50x 3 + 25x 2 -7x + 1
	5 1.898828 49.353681 54.93	x 4 -3x 3 + 5x 2 -3x + 1
	6 1.853006 59.360240	60	x 8 -5x 7 + 16x 6 -29x 5 + 37x 4 -29x 3 + 16x 2 -5x + 1
	7 1.732050	60	71.19	x 2 -x + 1
	8 1.626576 76.795630 81.02	x 4 -x 3 + 3x 2 -x + 1

  1.990903 [ 29.78, 40.42) then 0.951716 d .L(α) 0.447824 ≤ M(α) ≤ 0.215121 d .L(α)2.356599 [ 40.42, 48.68) then 0.826667 d .L(α) 0.599317 ≤ M(α) ≤ 0.128230 d .L(α)3.544544 [ 48.68, 54.93) then 0.820391 d .L(α) 0.524156 ≤ M(α) ≤ 0.197247 d .L(α) 3.016706 [ 54.93, 60) then 0.528810 d .L(α) 1.159859 ≤ M(α) ≤ 0.152149 d .L(α) 3.545630 [ 60, 71.19) then 0.952391 d .L(α) 0.434332 ≤ M(α) ≤ 0.547165 d .L(α) 1.677066 [ 71.19, 81.02) then 0.903804 d .L(α) 0.457938 ≤ M(α) ≤ 0.582441 d .L(α) 1.719218

Table 2 :

 2 P is the minimal polynomial of the algebraic integer α whose all conjugates lie in the sector [θ i , θ i ).

	[θ i , θ i ) [0, 18.26)	P x 3 -5x 2 + 7x -1	2 -d L(α) 1.75000	Our lower bound 4.59136	M(α) 6.22226	Our upper bound 7.43501	L(α) 14.0000
	[18.26,29.78)	x 3 -5x 2 + 8x -3	2.12500	3.18337	5.61347	11.4485	17.0000
	[29.78,40.42) x 4 -4x 3 + 7x 2 -4x + 1	1.06250	2.91779	4.33064	4.55268	17.0000
	[40.42,48.68)	x 3 -3x 2 + 4x -1	1.12500	2.10807	3.14790	5.08538	9.00000
	[48.68,54.93) x 4 -3x 3 + 5x 2 -3x + 1	0.81250	1.73766	2.96557	3.47122	13.0000
	[54.93,60)	x 3 -2x 2 + 3x -1	0.875000	1.41287	2.32472	3.4931	7.00000
	[60,71.19)	x 4 -2x 3 + 4x 2 -2x + 1	0.62500	2.23901	2.89005	4.26129	10.0000
	[71.19,81.02)