TY - GEN
T1 - Effective Hilbert’s Irreducibility Theorem for Primary Ideals
AU - Ishihara, Yuki
AU - Yokoyama, Kazuhiro
N1 - Publisher Copyright:
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2026.
PY - 2026
Y1 - 2026
N2 - Hilbert’s Irreducibility Theorem states that for a parametric irreducible polynomial f(A, X) of Q[A,X] with parameters A={a1,…,am} and indeterminates X={x1,…,xn}, the set Of={α∈Qm∣f(α,X)isirreducibleoverQ} forms a dense subset of Qm in the Euclidean topology, where Of is called a basic Hilbert subset w.r.t. f. We generalize this theorem to a prime or primary ideal P of Q[A,X] and propose an effective method to compute a Hilbert subset O in Qm such that P preserves its primality or primariness over O when P∩Q[A]={0}, i.e., there are no algebraic constraints between parameters. To explain more explicitly, we consider the specialization map φα:f(A,X)↦f(α,X) for α∈Qm. For a prime (primary) ideal P of Q[A,X] with P∩Q[A]={0}, our algorithm computes an irreducible polynomial f in Q[A,X] and a parametric ideal J of Q[A] such that φα(P) is a prime (primary) ideal for any α∈O=Of∩(Qm\VQ(J)), where VQ(J) denotes the set of zeros of J in Qm. In addition, our method can be applied to a primary ideal Q with Q∩Q[A]≠{0} if Q∩Q[A] is a prime ideal. In this case, φα(Q) is a primary ideal for any α∈Of∩(VQ(Q∩Q[A])\VQ(J)), which we call a semi-Hilbert subset for Q. We implement our algorithm on the computer algebra system Risa/Asir and present its applications including parametric primary decomposition.
AB - Hilbert’s Irreducibility Theorem states that for a parametric irreducible polynomial f(A, X) of Q[A,X] with parameters A={a1,…,am} and indeterminates X={x1,…,xn}, the set Of={α∈Qm∣f(α,X)isirreducibleoverQ} forms a dense subset of Qm in the Euclidean topology, where Of is called a basic Hilbert subset w.r.t. f. We generalize this theorem to a prime or primary ideal P of Q[A,X] and propose an effective method to compute a Hilbert subset O in Qm such that P preserves its primality or primariness over O when P∩Q[A]={0}, i.e., there are no algebraic constraints between parameters. To explain more explicitly, we consider the specialization map φα:f(A,X)↦f(α,X) for α∈Qm. For a prime (primary) ideal P of Q[A,X] with P∩Q[A]={0}, our algorithm computes an irreducible polynomial f in Q[A,X] and a parametric ideal J of Q[A] such that φα(P) is a prime (primary) ideal for any α∈O=Of∩(Qm\VQ(J)), where VQ(J) denotes the set of zeros of J in Qm. In addition, our method can be applied to a primary ideal Q with Q∩Q[A]≠{0} if Q∩Q[A] is a prime ideal. In this case, φα(Q) is a primary ideal for any α∈Of∩(VQ(Q∩Q[A])\VQ(J)), which we call a semi-Hilbert subset for Q. We implement our algorithm on the computer algebra system Risa/Asir and present its applications including parametric primary decomposition.
KW - Comprehensive Gröbner system
KW - Gröbner basis
KW - Hilbert’s irreducibility theorem
KW - Parametric ideal
KW - Primary ideal
UR - https://www.scopus.com/pages/publications/105021798997
U2 - 10.1007/978-3-032-09645-6_8
DO - 10.1007/978-3-032-09645-6_8
M3 - Conference contribution
AN - SCOPUS:105021798997
SN - 9783032096449
T3 - Lecture Notes in Computer Science
SP - 134
EP - 153
BT - Computer Algebra in Scientific Computing - 27th International Workshop, CASC 2025, Proceedings
A2 - Boulier, François
A2 - Mou, Chenqi
A2 - Sadykov, Timur M.
A2 - Vorozhtsov, Evgenii V.
PB - Springer Science and Business Media Deutschland GmbH
T2 - 27th International Workshop on Computer Algebra in Scientific Computing, CASC 2025
Y2 - 24 November 2025 through 28 November 2025
ER -