TY - JOUR
T1 - Generic Polynomials with Few Parameters
AU - Kemper, Gregor
AU - Mattig, Elena
N1 - Funding Information:
§The author gratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft. ¶E-mail: [email protected] ‖E-mail: [email protected]
PY - 2000/12
Y1 - 2000/12
N2 - We call a polynomial g(t1, . . . , tm,X ) over a field K generic for a group G if it has Galois group G as a polynomial inX , and if every Galois field extension N/L withK ⊆L and Gal(N/L) ≤G arises as the splitting field of a suitable specializationg (λ1, . . . , λm, X) withλi ∈L. We discuss how the rationality of the invariant field of a faithful linear representation leads to a generic polynomial which is often particularly simple and therefore useful. Then we consider various examples and applications in characteristic 0 and in positive characteristic. These include results on so-called vectorial polynomials and a generalization of an embedding criterion given by Abhyankar. We give recursive formulas for generic polynomials over a field of defining characteristic for the groups of upper unipotent and upper triangular matrices, and explicit formulae for generic polynomials for the groups GU2(q2) andGO3 (q).
AB - We call a polynomial g(t1, . . . , tm,X ) over a field K generic for a group G if it has Galois group G as a polynomial inX , and if every Galois field extension N/L withK ⊆L and Gal(N/L) ≤G arises as the splitting field of a suitable specializationg (λ1, . . . , λm, X) withλi ∈L. We discuss how the rationality of the invariant field of a faithful linear representation leads to a generic polynomial which is often particularly simple and therefore useful. Then we consider various examples and applications in characteristic 0 and in positive characteristic. These include results on so-called vectorial polynomials and a generalization of an embedding criterion given by Abhyankar. We give recursive formulas for generic polynomials over a field of defining characteristic for the groups of upper unipotent and upper triangular matrices, and explicit formulae for generic polynomials for the groups GU2(q2) andGO3 (q).
UR - http://www.scopus.com/inward/record.url?scp=0000308528&partnerID=8YFLogxK
U2 - 10.1006/jsco.1999.0385
DO - 10.1006/jsco.1999.0385
M3 - Article
AN - SCOPUS:0000308528
SN - 0747-7171
VL - 30
SP - 843
EP - 857
JO - Journal of Symbolic Computation
JF - Journal of Symbolic Computation
IS - 6
ER -