## Norm form equations. I.(English)Zbl 0578.10021

Let $$L$$ be an arbitrary finite extension of the rational function field $$k(z)$$ over an algebraically closed field $$k$$ of characteristic zero, let $$\mathcal O$$ denote the ring of elements of $$L$$ integral over $$k[z]$$, and let $$M$$ be a finitely generated $${\mathcal O}$$-module lying in a finite extension $$K$$ of $$L$$. Each valuation $$v$$ on $$K$$ is defined as the order of vanishing of the Laurent expansion in a local parameter $$z_ v$$, and those valuations with $$v(z)<0$$ are termed infinite. The height $$H(f)$$ of any element $$f$$ in $$K$$ is defined by $$H(f)=-\sum_{v}\min (0,v(f)),$$ and the height $$H({\mathcal F})$$ of any finite set $${\mathcal F}$$ of elements of $$K$$ is defined by $$H({\mathcal F})=-\sum_{v}\min (0,v(f); f\in {\mathcal F}).$$
The main results of this paper read as follows. Suppose that $$K$$ be a Galois extension of $$L$$. Then, for each $$c$$ in $$L$$ all the solutions $$x$$ in $$M$$ of the equation (*) $$\text{Norm}_{K/L}(x)=c$$ may be determined effectively. Moreover, these solutions are finite in number. Each solution $$x$$ in $$M$$ of (*) satisfies
$H(x)\leq (2d)^{3n-3} 2^{n^2} (H+H(c)+g_K+r_K+1),$
where $$H$$ denotes the height of an $$\mathcal O$$-basis $$x_1,\dots, x_n$$ of $$M$$, and $$d$$, $$g_K$$, $$r_K$$ denote the degree $$[K:L]$$, the genus of $$K/k$$, and the number of infinite valuations on $$K$$, respectively.

### MSC:

 11D57 Multiplicative and norm form equations 12J25 Non-Archimedean valued fields 11R58 Arithmetic theory of algebraic function fields
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### References:

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