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The constant term of the minimal polynomial of over ℚ
Fixed Point Theory and Applications volume 2013, Article number: 77 (2013)
Let be the Hecke group associated to for integer. In this paper, we determine the constant term of the minimal polynomial of denoted by .
The Hecke groups are defined to be the maximal discrete subgroups of generated by two linear fractional transformations
where λ is a fixed positive real number.
Hecke  showed that is Fuchsian if and only if for is an integer, or . In this paper, we only consider the former case and denote the corresponding Hecke groups by . It is well known that has a presentation as follows (see ):
These groups are isomorphic to the free product of two finite cyclic groups of orders 2 and q.
The first few Hecke groups are (the modular group), , , and . It is clear from the above that , but unlike in the modular group case (the case ), the inclusion is strict and the index is infinite as is discrete, whereas is not for .
On the other hand, it is well known that ζ, a primitive n th root of unity, satisfies the equation
In , Cangul studied the minimal polynomials of the real part of ζ, i.e., of over the rationals. He used a paper of Watkins and Zeitlin  to produce further results. Also, he made use of two classes of polynomials called Chebycheff and Dickson polynomials. It is known that for , the n th Chebycheff polynomial, denoted by , is defined by
Here we use Chebycheff polynomials.
For , Cangul denoted the minimal polynomial of over Q by . Then he obtained the following formula for the minimal polynomial .
Theorem 1 ([, Theorem 1])
Let and . Then
If , then , and if , then .
If m is an odd prime, then(5)
If , then(6)
If m is even and is odd, then(7)
Let m be odd and let p be a prime dividing m. If , then(8)
where . If , then
For the first four Hecke groups Γ, , , and , we can find the minimal polynomial, denoted by , of over Q as , , , and , respectively. However, for , the algebraic number is a root of a minimal polynomial of degree ≥3. Therefore, it is not possible to determine for as nicely as in the first four cases. Because of this, it is easy to find and study with the minimal polynomial of instead of itself. The minimal polynomial of has been used for many aspects in the literature (see [5–8] and ).
Notice that there is a relation
between and .
In , when the principal congruence subgroups of for prime were studied, we needed to know whether the minimal polynomial of is congruent to 0 modulo p for prime p and also the constant term of it modulo p.
In this paper, we determine the constant term of the minimal polynomial of . We deal with odd and even q cases separately. Of course, this problem is easier to solve when q is odd.
2 The constant term of
In this section, we calculate the constant term for all values of q. Let c denote the constant term of the minimal polynomial of , i.e.,
We know from [, Lemma, p.473] that the roots of are with , h odd and . Being the constant term, c is equal to the product of all roots of :
Therefore we need to calculate the product on the right-hand side of (11). To do this, we need the following result given in .
Lemma 1 .
We now want to obtain a similar formula for cosine. Replacing θ by , we get
Let now μ denote the Möbius function defined by
for . It is known that
Using this last fact, we obtain
Finally, as , we can write (16) as
Note that if q is even, then
while if q is odd, then
as . Also note that
To compute c, we let in (17). If d is odd, then as by (20). So, we are only concerned with even d. Indeed, if q is odd, then the left-hand side at is equal to ±1. Therefore we have the following result.
Theorem 2 Let q be odd. Then
Proof It follows from (19) and (20). □
Let us now investigate the case of even q. As , h must be odd. So, by a similar discussion, we get the following.
Theorem 3 Let q be even. Then
Proof Note that by (20), the right-hand side of (22) becomes a product of ’s and ’s. Above we saw that we can omit the former ones as they tend to ±1 as θ tends to 0. Now, as , there are equal numbers of the latter kind factors in the numerator and denominator, i.e., if there is a factor in the numerator, then there is a factor in the denominator. Then using the fact that
we can calculate c.
In fact the calculations show that there are three possibilities:
(i) Let , . Then the only divisors of q such that are and . Therefore
(ii) Secondly, let , , p odd prime. Then the only divisors of q such that are , , and . Therefore
(iii) Let q be different from above. Then q can be written as
where are distinct odd primes and , .
Here we consider the first two cases and .
Let , i.e., let . We have already discussed the case . Let . Then the only divisors d of q with are , , and . Therefore
Now let , i.e., let , (). Similarly, all divisors d of q such that are , , , , , , and . Therefore
Finally, , i.e., let
In this case the proof is similar, but rather more complicated. In fact, the number of all divisors d of q such that is . There is divisor of the form
There are divisors of the form
There are divisors of the form
If we continue, we can find other divisors d of q, similarly. Finally, there is divisor of the form . Thus, the product of all coefficients d in the factors in the numerator is equal to the product of all coefficients e in the factors in the denominator implying . Therefore the proof is completed. □
Now we give an example for all possible even q cases.
Example 1 (i) Let . The only divisors of 8 such that are and 4. Therefore
(ii) Let . The only divisors of 14 such that are and 1. Therefore
(iii) Let . The only divisors of 24 such that are and 4. Therefore
(iv) Let . The only divisors of 30 such that are and 1. Therefore
Hecke E: Über die bestimmung dirichletscher reihen durch ihre funktionalgleichungen. Math. Ann. 1936, 112: 664–699. 10.1007/BF01565437
Cangul IN, Singerman D: Normal subgroups of Hecke groups and regular maps. Math. Proc. Camb. Philos. Soc. 1998, 123: 59–74. 10.1017/S0305004197002004
Cangul IN:The minimal polynomials of over ℚ. Probl. Mat. - Wyż. Szk. Pedagog. Bydg. 1997, 15: 57–62.
Watkins W, Zeitlin J:The minimal polynomial of . Am. Math. Mon. 1993, 100(5):471–474. 10.2307/2324301
Arnoux P, Schmidt TA: Veech surfaces with non-periodic directions in the trace field. J. Mod. Dyn. 2009, 3(4):611–629.
Beslin S, De Angelis V:The minimal polynomials of and . Math. Mag. 2004, 77(2):146–149.
Rosen R, Towse C: Continued fraction representations of units associated with certain Hecke groups. Arch. Math. 2001, 77(4):294–302. 10.1007/PL00000494
Schmidt TA, Smith KM: Galois orbits of principal congruence Hecke curves. J. Lond. Math. Soc. 2003, 67(3):673–685. 10.1112/S0024610703004113
Surowski D, McCombs P:Homogeneous polynomials and the minimal polynomial of . Mo. J. Math. Sci. (Print) 2003, 15(1):4–14.
Ikikardes S, Sahin R, Cangul IN: Principal congruence subgroups of the Hecke groups and related results. Bull. Braz. Math. Soc. 2009, 40(4):479–494. 10.1007/s00574-009-0023-y
Keng HL, Yuan W: Applications of Number Theory to Numerical Analysis. Springer, Berlin; 1981.
Dedicated to Professor Hari M Srivastava.
Both authors are supported by the Scientific Research Fund of Uludag University under the project number F2012/15 and the second author is supported under F2012/19.
The authors declare that they have no competing interests.
The authors completed the paper alone and they read and approved the final manuscript.
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Demirci, M., Cangül, I.N. The constant term of the minimal polynomial of over ℚ. Fixed Point Theory Appl 2013, 77 (2013). https://doi.org/10.1186/1687-1812-2013-77
- Hecke groups
- minimal polynomial
- constant term