Hilbert’s Irreducibility Theorem states that for a parametric irreducible polynomial f(A, X) of \(\mathbb {Q}[A,X]\) with parameters \(A=\{a_1,\ldots ,a_m\}\) and indeterminates \(X=\{x_1,\ldots ,x_n\}\) , the set \(\mathcal {O}_f=\{\alpha \in \mathbb {Q}^m \mid f(\alpha ,X) \text { is irreducible over }\mathbb {Q}\}\) forms a dense subset of \(\mathbb {Q}^m\) in the Euclidean topology, where \(\mathcal {O}_f\) is called a basic Hilbert subset w.r.t. f. We generalize this theorem to a prime or primary ideal P of \(\mathbb {Q}[A,X]\) and propose an effective method to compute a Hilbert subset \(\mathcal {O}\) in \(\mathbb {Q}^m\) such that P preserves its primality or primariness over \(\mathcal {O}\) when \(P\cap \mathbb {Q}[A]=\{0\}\) , i.e., there are no algebraic constraints between parameters. To explain more explicitly, we consider the specialization map \(\varphi _\alpha :f(A,X)\mapsto f(\alpha ,X)\) for \(\alpha \in \mathbb {Q}^m\) . For a prime (primary) ideal P of \(\mathbb {Q}[A,X]\) with \(P\cap \mathbb {Q}[A]=\{0\}\) , our algorithm computes an irreducible polynomial f in \(\mathbb {Q}[A,X]\) and a parametric ideal J of \(\mathbb {Q}[A]\) such that \(\varphi _\alpha (P)\) is a prime (primary) ideal for any \(\alpha \in \mathcal {O}=\mathcal {O}_f\cap (\mathbb {Q}^m\setminus V_{\mathbb {Q}}(J))\) , where \(V_{\mathbb {Q}}(J)\) denotes the set of zeros of J in \(\mathbb {Q}^m\) . In addition, our method can be applied to a primary ideal Q with \(Q\cap \mathbb {Q}[A]\not =\{0\}\) if \(Q\cap \mathbb {Q}[A]\) is a prime ideal. In this case, \(\varphi _\alpha (Q)\) is a primary ideal for any \(\alpha \in \mathcal {O}_f\cap (V_{\mathbb {Q}}(Q\cap \mathbb {Q}[A])\setminus V_{\mathbb {Q}}(J))\) , 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.

Effective Hilbert’s Irreducibility Theorem for Primary Ideals

会议论文
发表日期：

2026年

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Yuki Ishihara,
Kazuhiro Yokoyama

摘要

Hilbert’s Irreducibility Theorem states that for a parametric irreducible polynomial f(A, X) of \(\mathbb {Q}[A,X]\) with parameters \(A=\{a_1,\ldots ,a_m\}\) and indeterminates \(X=\{x_1,\ldots ,x_n\}\) , the set \(\mathcal {O}_f=\{\alpha \in \mathbb {Q}^m \mid f(\alpha ,X) \text { is irreducible over }\mathbb {Q}\}\) forms a dense subset of \(\mathbb {Q}^m\) in the Euclidean topology, where \(\mathcal {O}_f\) is called a basic Hilbert subset w.r.t. f. We generalize this theorem to a prime or primary ideal P of \(\mathbb {Q}[A,X]\) and propose an effective method to compute a Hilbert subset \(\mathcal {O}\) in \(\mathbb {Q}^m\) such that P preserves its primality or primariness over \(\mathcal {O}\) when \(P\cap \mathbb {Q}[A]=\{0\}\) , i.e., there are no algebraic constraints between parameters. To explain more explicitly, we consider the specialization map \(\varphi _\alpha :f(A,X)\mapsto f(\alpha ,X)\) for \(\alpha \in \mathbb {Q}^m\) . For a prime (primary) ideal P of \(\mathbb {Q}[A,X]\) with \(P\cap \mathbb {Q}[A]=\{0\}\) , our algorithm computes an irreducible polynomial f in \(\mathbb {Q}[A,X]\) and a parametric ideal J of \(\mathbb {Q}[A]\) such that \(\varphi _\alpha (P)\) is a prime (primary) ideal for any \(\alpha \in \mathcal {O}=\mathcal {O}_f\cap (\mathbb {Q}^m\setminus V_{\mathbb {Q}}(J))\) , where \(V_{\mathbb {Q}}(J)\) denotes the set of zeros of J in \(\mathbb {Q}^m\) . In addition, our method can be applied to a primary ideal Q with \(Q\cap \mathbb {Q}[A]\not =\{0\}\) if \(Q\cap \mathbb {Q}[A]\) is a prime ideal. In this case, \(\varphi _\alpha (Q)\) is a primary ideal for any \(\alpha \in \mathcal {O}_f\cap (V_{\mathbb {Q}}(Q\cap \mathbb {Q}[A])\setminus V_{\mathbb {Q}}(J))\) , 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.