Galois Theory of Reciprocal Polynomials

Viana, P. A.; Veloso, Paula M.

doi:10.2307/2695648

Cited by 4 publications

(6 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The roots of such a polynomial come in pairs. The Galois group of the number field defined by the polynomial is a subgroup of the semidirect product (Z/2Z) g ⋊ S g where S g is the symmetric group in g elements (see [VV02] for a nice account on this classical fact), and S g acts on (Z/2Z) g by permutation of the factors.…”

Section: Galois Groupsmentioning

confidence: 99%

“…Its roots define a number field K of degree at most 2g over Q which is a quadratic extension of the so-called trace field of A. The Galois group of K is isomorphic to a subgroup of the semi-direct product (Z/2Z) g ⋊ Σ g where Σ g is the symmetric group in g variables (see [VV02] for details). The field K and the Galois group only depend on the conjugacy class of A.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Typical and atypical properties of periodic Teichmueller geodesics

Hamenstaedt

2014

Preprint

View full text Add to dashboard Cite

Consider a component Q of a stratum in the moduli space of area one abelian differentials on a surface of genus g. Call a property P for periodic orbits of the Teichmüller flow on Q typical if the growth rate of orbits with property P is maximal. Typical are: The logarithms of the eigenvalues of the symplectic matrix defined by the orbit are arbitrarily close to the Lyapunov exponents of Q, and its trace field is a totally real splitting field of degree g over Q. If g ≥ 3 then periodic orbits whose SL(2, R)-orbit closure equals Q are typical. We also show that Q contains only finitely many algebraically primitive Teichmüller curves, and only finitely many affine invariant submanifolds of rank ℓ ≥ 2.

show abstract

Section: Galois Groupsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Typical and atypical properties of periodic Teichmueller geodesics

Hamenstaedt

2014

Preprint

View full text Add to dashboard Cite

show abstract

“…To a reciprocal polymial f we can associate the trace polynomial F of half the degree, by writing f in terms of the variable z = x + 1/x (constructing the trace polynomial is a simple matter of linear algebra, which we leave to the reader). While the Galois group G( f ) of a reciprocal polynomial f (of degree 2n now) is a subgroup of the hyperoctahedral group C 2 ≀ S n , the Galois group of the associated trace polynomial F is the image of G( f ) under the natural projection to S n -see [45] for a very accessible introduction to all of the above.…”

Section: A Bit About Polynomialsmentioning

confidence: 99%

Large Galois groups with applications to Zariski density

Rivin

2013

Preprint

View full text Add to dashboard Cite

We introduce the first provably efficient algorithm to check if a finitely generated subgroup of an almost simple semi-simple group over the rationals is Zariski-dense. We reduce this question to one of computing Galois groups, and to this end we describe efficient algorithms to check if the Galois group of a polynomial p with integer coefficients is "generic" (which, for arbitrary polynomials of degree n means the full symmetric group S n , while for reciprocal polynomials of degree 2n it means the hyperoctahedral group C 2 ≀S n .). We give efficient algorithms to verify that a polynomial has Galois group S n , and that a reciprocal polynomial has Galois group C 2 ≀S n . We show how these algorithms give efficient algorithms to check if a set of matrices G in SL(n, Z) or Sp(2n, Z) generate a Zariski dense subgroup. The complexity of doing this inSL(n, Z) is of order O(n 4 log n log G ) log ǫ and in Sp(2n, Z) the complexity is of order O(n 8 log n log G ) log ǫ In general semisimple groups we show that Zariski density can be confirmed or denied in time of order O(n 1 4 log G log ǫ), where ǫ is the probability of a wrong "NO" answer, while G is the measure of complexity of the input (the maximum of the Frobenius norms of the generating matrices). The algorithms work essentially without change over algebraic number fields, and in other semi-simple groups. However, we restrict to the case of the special linear and symplectic groups and rational coefficient in the interest of clarity.

show abstract

“…For some reciprocal polynomials f (x) the extension of N by G g splits, and for some it does not. For example, if f (x) = n (x) is the nth cyclotomic polynomial then in [6] it is shown that the extension splits if and only if either 4 | n or p | n for some prime p with p ≡ 3 (mod 4). In this cyclotomic case N = C 2 is cyclic of order 2, generated by complex conjugation.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, N is abelian. The generic case (treated in [6]) has G g % S d (the symmetric group acting on the d roots of g) and N % C d 2 . In this paper we shall consider the case where f (x) is a Salem polynomial, the minimal polynomial of a Salem number.…”

Section: Introductionmentioning

confidence: 99%

Galois theory of Salem polynomials

Christopoulos¹,

McKee²

2009

Math. Proc. Camb. Phil. Soc.

View full text Add to dashboard Cite

Let f(x) ∈ [x] be a monic irreducible reciprocal polynomial of degree 2d with roots r1, 1/r1, r2, 1/r2, . . ., rd, 1/rd. The corresponding trace polynomial g(x) of degree d is the polynomial whose roots are r1 + 1/r1, . . ., rd + 1/rd. If the Galois groups of f and g are Gf and Gg respectively, then Gg ≅ Gf/N, where N is isomorphic to a subgroup of C2d. In a naive sense, the generic case is Gf ≅ C2d ⋊ Sd, with N ≅ C2d and Gg ≅ Sd. When f(x) has extra structure this may be reflected in the Galois group, and it is not always true even that Gf ≅ N ⋊ Gg. For example, for cyclotomic polynomials f(x) = Φn(x) it is known that Gf ≅ N ⋊ Gg if and only if n is divisible either by 4 or by some prime congruent to 3 modulo 4.In this paper we deal with irreducible reciprocal monic polynomials f(x) ∈ [x] that are ‘close’ to being cyclotomic, in that there is one pair of real positive reciprocal roots and all other roots lie on the unit circle. With the further restriction that f(x) has degree at least 4, this means that f(x) is the minimal polynomial of a Salem number. We show that in this case one always has Gf ≅ N ⋊ Gg, and moreover that N ≅ C2d or C2d−1, with the latter only possible if d is odd.

show abstract

Galois Theory of Reciprocal Polynomials

Cited by 4 publications

References 0 publications

Typical and atypical properties of periodic Teichmueller geodesics

Typical and atypical properties of periodic Teichmueller geodesics

Large Galois groups with applications to Zariski density

Galois theory of Salem polynomials

Contact Info

Product

Resources

About