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Number of solutions of the homogeneous norm form equation. (Lösungsanzahl der homogenen Normformengleichung.) (German) Zbl 0820.11020
In 1972 W. M. Schmidt [Ann. Math., II. Ser. 96, 526-551 (1972; Zbl 0245.10008)] characterized the norm form equations with only finitely many solutions. In the present paper the author shows that under certain conditions, for almost all primes $$p$$, the norm form equation with right hand side $$p$$ has, up to trivial cases, at most one solution. Moreover, under certain assumptions, than upper bound is given for the number of solutions of general norm form equations.
The proof uses and extends some earlier results of the author [Compos. Math. 88, 25-38 (1993; Zbl 0782.11011)] based mainly on the theorem of Evertse-Laurent-van der Poorten on $$S$$-unit equations.
Reviewer: I.Gaál (Debrecen)

##### MSC:
 11D57 Multiplicative and norm form equations 11D75 Diophantine inequalities
##### Keywords:
norm form equations; upper bound; $$S$$-unit equations
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##### References:
 [1] Evertse, J.H. , On sums of S-units and linear recurrences , Compos. Math. 53 (1984), 225-244. · Zbl 0547.10008 · numdam:CM_1984__53_2_225_0 · eudml:89685 [2] Langmann, K. , Picard-Borel-Räume , Math. Ann, 284 (1989), 138-160. · Zbl 0647.32024 · doi:10.1007/BF01443510 · eudml:164545 [3] Langmann, K. , Eindeutigkeit der Lösungen der Gleichung xd + yd = ap , Compos. Math. 88 (1993), 25-38. · Zbl 0782.11011 · numdam:CM_1993__88_1_25_0 · eudml:90239 [4] Langmann, K. , Picardindex und ganzalgebraische Punkte , Math. Ann. 291 (1991), 663-690. · Zbl 0724.14012 · doi:10.1007/BF01445233 · eudml:164892 [5] Schmidt, W.M. , Norm form equations , Annals of Math. 96 (1972), 526-551. · Zbl 0226.10024 · doi:10.2307/1970824
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