On the number of pairs of polynomials with given resultant or given semi- resultant.

*(English)*Zbl 0798.11043Let \(\mathbb{K}\) be a finitely generated extension field of \(\mathbb{Q}\), \(G\) a finite normal extension of \(\mathbb{K}\), and \(A\) an integrally closed subring of \(K\) with 1 which has \(\mathbb{K}\) as its quotient field and which is finitely generated over \(\mathbb{Z}\). Denote by \(A^*\) the set of invertible elements of \(A\).

Let \(f(x)\) and \(g(x)\) be monic polynomials in \(\mathbb{K} [x]\) with zeros \(\alpha_ 1, \dots, \alpha_ m\) and \(\beta_ 1, \dots, \beta_ n\) respectively. The semi-resultant \(R^* (f,g)\) of \(f(x_ i)\) and \(g(x)\) is defined by \[ R^* (f,g) = \prod{'}_{i,j} (\alpha_ i - \beta_ j), \] where the product \(\prod'\) is over all pairs \(i,j\) such that \(\alpha_ i - \beta_ j \neq 0\).

For given \(a \in A\) it is considered here the semi-resultant equation (1) \(R^*(f,g) \in aA^*\) in monic \(f(x)\), \(g(x) \in A[x]\) having all their zeros in \(G\). The pairs \((f,g)\) and \((f',g')\) of monic polynomials in \(A[x]\) are called weakly \(A\)-equivalent if \(f'(x) = \varepsilon^{\deg(f)} f(x/ \varepsilon + b)\) and \(g'(x) = \varepsilon^{\deg(g)} g(x/ \varepsilon + b)\) for some \(\varepsilon \in A^*\) and \(b \in A\). For \(f(x) \in K[x]\), \(\omega(f)\) denotes the number of distinct zeros of \(f\).

The main result of this paper is that if \((f,g)\) is a solution of (1) such that \(\omega(f)\), \(\omega (g) \geq 2\) and if \(f(x)\) and \(g(x)\) have no common zero then \(\omega (f) + \omega (g) \geq 5\), then \(\omega (f) + \omega (g) \leq 24 \cdot 7^{D(3s + 2t)}\), where \(D,s\) and \(t\) are constants depending only on \(\mathbb{K}\) and \(G\).

In the second theorem an explicit upper bound for the number of weak \(A\)- equivalence classes of solutions \((f,g)\) of (1) is given under the further restriction that the degree of \(f\) and \(g\) is fixed.

The author considered the special case, when \(\mathbb{K}\) is an algebraic number field, too.

Let \(f(x)\) and \(g(x)\) be monic polynomials in \(\mathbb{K} [x]\) with zeros \(\alpha_ 1, \dots, \alpha_ m\) and \(\beta_ 1, \dots, \beta_ n\) respectively. The semi-resultant \(R^* (f,g)\) of \(f(x_ i)\) and \(g(x)\) is defined by \[ R^* (f,g) = \prod{'}_{i,j} (\alpha_ i - \beta_ j), \] where the product \(\prod'\) is over all pairs \(i,j\) such that \(\alpha_ i - \beta_ j \neq 0\).

For given \(a \in A\) it is considered here the semi-resultant equation (1) \(R^*(f,g) \in aA^*\) in monic \(f(x)\), \(g(x) \in A[x]\) having all their zeros in \(G\). The pairs \((f,g)\) and \((f',g')\) of monic polynomials in \(A[x]\) are called weakly \(A\)-equivalent if \(f'(x) = \varepsilon^{\deg(f)} f(x/ \varepsilon + b)\) and \(g'(x) = \varepsilon^{\deg(g)} g(x/ \varepsilon + b)\) for some \(\varepsilon \in A^*\) and \(b \in A\). For \(f(x) \in K[x]\), \(\omega(f)\) denotes the number of distinct zeros of \(f\).

The main result of this paper is that if \((f,g)\) is a solution of (1) such that \(\omega(f)\), \(\omega (g) \geq 2\) and if \(f(x)\) and \(g(x)\) have no common zero then \(\omega (f) + \omega (g) \geq 5\), then \(\omega (f) + \omega (g) \leq 24 \cdot 7^{D(3s + 2t)}\), where \(D,s\) and \(t\) are constants depending only on \(\mathbb{K}\) and \(G\).

In the second theorem an explicit upper bound for the number of weak \(A\)- equivalence classes of solutions \((f,g)\) of (1) is given under the further restriction that the degree of \(f\) and \(g\) is fixed.

The author considered the special case, when \(\mathbb{K}\) is an algebraic number field, too.

Reviewer: A.Pethö (Debrecen)