## Cyclotomic Points on Curves

Citations: | 3 - 0 self |

### BibTeX

@MISC{Beukers_cyclotomicpoints,

author = {F. Beukers and C. J. Smyth},

title = {Cyclotomic Points on Curves},

year = {}

}

### OpenURL

### Abstract

. We show that a plane algebraic curve f = 0 over the complex numbers has on it either at most 22V (f) points whose coordinates are both roots of unity, or infinitely many such points. Here V (f) is the area of the Newton polytope of f. We present an algorithm for finding all these points. 1. Introduction. When does a curve f(x, y) = 0 with complex coe#cients have cyclotomic points on it? By cyclotomic points we mean points (x, y) with x and y both roots of unity. How do we estimate the number of cyclotomic points on a given curve? How do we actually find all these points? In Section 2 we present an algorithm for finding the cyclotomic part of a polynomial in one variable. We do this first for polynomials with rational coe#cients, and then show how the algorithm can be extended to polynomials with complex coe#cients. In Section 3 we give an algorithm for finding all the cyclotomic points on a curve. This algorithm uses the algorithm of Section 2. Again, we first present an algorithm ...

### Citations

826 |
Introduction to toric varieties
- Fulton
- 1993
(Show Context)
Citation Context ...for some positive integer k, then the number of such points is at most 2kV (f). 8 F. BEUKERS AND C.J. SMYTH Proof. The first statement of the lemma is a based on a slight strengthening, due to Fulton =-=[F]-=- p. 122, of a result of D. N. Bernstein, A.G. Kouchnirenko, and A.G. Khovanskii which, in the two variable case, gives an upper bound for the number of points of C #2 on two plane algebraic curves f =... |

175 |
Integral Matrices
- Newman
- 1972
(Show Context)
Citation Context ...x ai+bj y ci+dj , and so on pairs of monomials by this action on each coordinate. Then we can write the relations connecting x, y with u, v as x y A = u v . Now, putting A into Smith Normal Form ([N], p. 26) yields two matrices U and W in SL 2 (Z) with WAU = D say, where D = diag(d 1 , d 2 ), and d 1 and d 2 are positive integers with d 1 |d 2 . Also d 1 d 2 = I . Hence x y W -1 D = u v U ... |

72 |
Speculations concerning the range of Mahler’s measure
- Boyd
- 1981
(Show Context)
Citation Context ...pect its Mahler measure (the geometric mean of |f | on |x| = |y| = 1) to be small. And indeed it is the twovariable polynomial with integer coe#cients of smallest known Mahler measure greater than 1 (=-=[B]-=-). Note that the primes 2, 3, 5 and 7 all appear as divisors of the orders of the coordinates of cyclotomic points on this curve. In fact, this must happen for any f having more than 14V (f) cyclotomi... |

60 |
On a question of Lehmer and the number of irreducible factors of a polynomial
- Dobrowolski
(Show Context)
Citation Context ...first part of (ii) was stated without proof in [Sm], where it was used to find the cyclotomic factors of a family of polynomials. The converse part of (ii) is a special case of a result of Dobrowolski=-=[D]-=-, Lemma 2(i). To prove (i), suppose that # is any zero of g. Then so is one of # 2 , and hence so is one of # 4 , one of # 8 , and so on. As g has only finitely many zeroes, two of these powers must b... |

46 |
Diophantine geometry. An introduction
- Hindry, Silverman
- 2000
(Show Context)
Citation Context ...ating conjectures of Mordell and Manin--Mumford, describes the intersections of a semi-abelian variety A over C with the division group of a finitely generated subgroup of A. See Hindry and Silverman =-=[HS]-=-, especially p.439, for a very readable, up-to-date account of results in this area. 4.2 Lemmas for the proof. For the proof of the theorem, we need the following two lemmas. The first one is a partic... |

33 |
Division points on curves
- Lang
- 1965
(Show Context)
Citation Context ...btorus. Our result contains an alternative proof of the special case when the finitely generated group is the trivial group. This was in fact first proved by Ihara, Serre and Tate, independently (see =-=[La]-=-). Quantitative (though large) bounds on the number of such points, when this number is finite, follow from a general result of Schmidt [Sc] on the number of maximal torsion cosets on a variety. Our t... |

32 |
Division points on semi-abelian varieties
- McQuillan
- 1995
(Show Context)
Citation Context ...c] on the number of maximal torsion cosets on a variety. Our theorem gives an almost sharp quantative upper bound in the curve case. A far more general conjecture of Lang, proved in 1995 by McQuillan =-=[M]-=-, incorporating conjectures of Mordell and Manin--Mumford, describes the intersections of a semi-abelian variety A over C with the division group of a finitely generated subgroup of A. See Hindry and ... |

24 |
The difference body of a convex
- Rogers, Shephard
- 1957
(Show Context)
Citation Context ...he plane, its di#erence body has area 6 6V (S), with equality i# S is a triangle. [In n dimensions the corresponding bound is 2n n V (S), with equality i# S is an n-simplex. See Rogers and Shephard [=-=RS]-=- for a surprisingly simple proof of this general upper bound.] Thus as V (f) = V (f + ), we have 2V (f, f + ) 6 (6 - 1 - 1)V (f) = 4V (f), and so, by Lemma 2, f has at most 4V (f) cyclotomic points. 4... |

24 |
Polynomial equations and convex polytopes
- Sturmfels
- 1998
(Show Context)
Citation Context ...mixed volumes is discussed in [E], p.82, but for our purposes we can just use the fact that 2V (f, g) = V (N (f) +N (g)) - V (f) - V (g). (A very accessible account of this kind of result is given in =-=[St]-=-.) For the second part, we have N (g) = kN (f), so that V (N (f) +N (g)) - V (f) - V (g) = ((1 + k) 2 - 1 - k 2 )V (f) = 2kV (f). Lemma 3. (i) If B 1 , , B k are convex bodies in R n , then V (B 1 + +... |

8 |
Jing Ping Wang. One symmetry does not imply integrability
- Beukers, Sanders
- 1998
(Show Context)
Citation Context ... extra solutions. In particular, when n # 1 (mod 6) we have at least 20n solutions. Since V (f) = 2n we have at least 10V (f) solutions in this case. 5.5 Application to generalised Lie-symmetries. In =-=[BSW]-=- the authors study generalised Lie-symmetries of certain partial evolution equations. It turned out that some systems do allow for infinitely many symmetries if a particular diophantine equation in ro... |

5 | numbers of negative trace
- Smyth, Salem
(Show Context)
Citation Context ... irreducible polynomial with integer coefficients. The obvious example α = √ 2 shows that α and −α can be conjugate without α being a root of unity. The first part of (ii) was stated without proof in =-=[Sm]-=-, where it was used to find the cyclotomic factors of a family of polynomials. The converse part of (ii) is a special case of a result of Dobrowolski[D], Lemma 2(i). To prove (i), suppose that α is an... |

4 |
Effective tests for cyclotomic polynomials, Symbolic and Algebraic Computation
- Bradford, Davenport
- 1988
(Show Context)
Citation Context ...part of f, which is (Cf)(x) = Y f(#)=0 # root of 1 (x - #). An algorithm for finding the cyclotomic part of a polynomial, using essentially the same ideas, was given earlier by Bradford and Davenport =-=[BD]-=-. First of all, we can clearly assume that f is monic and not divisible by x. Define Gf by (Gf)(x) = gcd((f(x), f(x 2 )f(-x 2 )). Then we claim that Cf is given recursively in pseudocode by Function C... |

4 | The Lehmer constants of an annulus
- Dubickas, Smyth
(Show Context)
Citation Context ...f the second kind, defined by Un (z + z -1 ) = z n+1 - z -(n+1) z - z -1 . Proposition. The polynomial U # n (X) has no zeroes of the form 2 cos 2#q #= 0 for any rational q. This result is applied in =-=[DS]-=-. Curiously, as remarked there, this result is in contrast to the situation for the Chebyshev polynomials Tn (X) of the first kind. For as T # n (X) = nUn-1 (X), all zeroes of T # n (X) are of the for... |

4 |
numbers of negative trace
- Salem
(Show Context)
Citation Context ...me irreducible polynomial with integer coe#cients. The obvious example # = # 2 shows that # and -# can be conjugate without # being a root of unity. The first part of (ii) was stated without proof in =-=[Sm]-=-, where it was used to find the cyclotomic factors of a family of polynomials. The converse part of (ii) is a special case of a result of Dobrowolski[D], Lemma 2(i). To prove (i), suppose that # is an... |

3 |
une conjecture de Serge Lang, Astérisque 24–25
- Liardet, Sur
- 1975
(Show Context)
Citation Context ... of Q ab , then f has at most 2V (f) cyclotomic points. Finally, the constant 22 above cannot be replaced by any constant smaller than 16. The constants 4 and 2 are best possible. A result of Liardet =-=[Li]-=- tells us that if a plane curve over C has infinite intersection with the division group of a finitely generated multiplicative group, then the curve has an irreducible component which is the translat... |

3 |
Heights of points on subvarieties of G n m, Number theory
- Schmidt
- 1996
(Show Context)
Citation Context ...t first proved by Ihara, Serre and Tate, independently (see [La]). Quantitative (though large) bounds on the number of such points, when this number is finite, follow from a general result of Schmidt =-=[Sc]-=- on the number of maximal torsion cosets on a variety. Our theorem gives an almost sharp quantative upper bound in the curve case. A far more general conjecture of Lang, proved in 1995 by McQuillan [M... |

2 |
Uber den Vektorenbereich eines konvexen ebenen Bereiches
- Rademacher
- 1925
(Show Context)
Citation Context ...f f + is a 180 # rotation of that of f. Thus the Minkowski sum N (f)+N (f + ) is what is called the di#erence body of N (f), namely the set {x 1 -x 2 | x 1 , x 2 # N (f)}. It was proved by Rademacher =-=[R-=-] that for any convex body S in the plane, its di#erence body has area 6 6V (S), with equality i# S is a triangle. [In n dimensions the corresponding bound is 2n n V (S), with equality i# S is an n-s... |

1 |
Heights of points on subvarieties ofG n m , Number theory
- Schmidt
- 1996
(Show Context)
Citation Context ...t first proved by Ihara, Serre and Tate, independently (see [La]). Quantitative (though large) bounds on the number of such points, when this number is finite, follow from a general result of Schmidt =-=[Sc]-=- on the number of maximal torsion cosets on a variety. Our theorem gives an almost sharp quantative upper bound in the curve case. A far more general conjecture of Lang, proved in 1995 by McQuillan [M... |