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On the number of pairs of polynomials with given resultant or given semi- resultant. (English) Zbl 0798.11043
Let $$\mathbb{K}$$ 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.

##### MSC:
 11R09 Polynomials (irreducibility, etc.) 11D57 Multiplicative and norm form equations