Vaserstein, L. N.; Wheland, E. R. Vanishing polynomial sums. (English) Zbl 1073.12001 Commun. Algebra 31, No. 2, 751-772 (2003). The main result of the paper gives an improved version of the generalization of the abc-theorem for function fields of zero characteristic. Precisely, if \(y_{1}+\dots+y_{m}=y_{0}\), where \(m\geq 2\), \(y_{i} \in F[t]\) (\(i=0,\dots,m\)), \(\text{char}(F)=0\), \(\gcd(y_{1},\dots,y_{m})=1\), not all \(y_{i}\) are constant, and no subsum of \(y_{1}\), \(\dots\), \(y_{m}\) vanishes, then \(\deg(y_{0}) < (m-1)\sum_{i=0}^{m}\nu(y_{i})\), and \(\deg(y_{0}) \leq (\nu(y_{0}\dots y_{m})-1)m(m-1)/2\). This improves on known previous results by W. D. Brownawell and D. W. Masser [Math. Proc. Camb. Philos. Soc. 100, 427–434 (1986; Zbl 0612.10010)].The result is applied to the generalized Fermat equation, \(c_{0}x_{0}^{n_{0}}+\dots+c_{m}x_{m}^{n_{m}}=0\), in \(m+1\) unknowns \(x_{i}\) in the polynomials ring \(F[t]\), with \(\text{char}(F)=0\), \(n\geq 2\), and \(c_{i}\in F\setminus\{0\}\). The authors prove that such equation has no non–trivial solutions when \(\sum 1/n_{i} \leq 1/(m-1)\), improving on a previous result by D. J. Newman and M. Slater [J. Number Theory 11, 477–487 (1979; Zbl 0407.10039)]. Reviewer: Leonardo Cangelmi (Pescara) Cited in 1 ReviewCited in 1 Document MSC: 12E05 Polynomials in general fields (irreducibility, etc.) 11D41 Higher degree equations; Fermat’s equation Keywords:abc-theorem; vanishing polynomial sums; generalized Fermat equation Citations:Zbl 0612.10010; Zbl 0407.10039 PDFBibTeX XMLCite \textit{L. N. Vaserstein} and \textit{E. R. Wheland}, Commun. Algebra 31, No. 2, 751--772 (2003; Zbl 1073.12001) Full Text: DOI References: [1] Beukers F., Duke Math. J. 91 pp 61– (1998) · Zbl 1038.11505 [2] Beukers F., Acta Arith. 78 pp 189– (1966) [3] Brownwell D., Math. Proc. Cambridge Philos. Soc. 100 pp 427– (1986) · Zbl 0612.10010 [4] Browkin J., Math. Comp. 62 pp 931– (1994) [5] Darmon H., Bull. London Math. Soc. 27 pp 513– (1995) · Zbl 0838.11023 [6] Elkies N.D., Math. Computation 51 pp 825– (1988) [7] García A., Manuscripta Math. 59 pp 457– (1987) · Zbl 0637.12015 [8] Kaplansky I., An Introduction to Differential Algebra (1957) · Zbl 0083.03301 [9] Lander L.J., Bull. Amer. Math. Soc. 72 pp 1079– (1996) · Zbl 0145.04903 [10] Lang S., Math Talks for Undergraduates (1999) · Zbl 0923.00001 [11] Mason R.C., I. J. of Number Theory 22 pp 190– (1986) · Zbl 0578.10021 [12] Mueller J., Acta Arithmetica 1 pp 7– (1993) [13] Mueller J., Bull. London Math. Soc. 32 pp 163– (2000) · Zbl 1043.11035 [14] Newman D.J., J. Number Theory 11 pp 477– (1979) · Zbl 0407.10039 [15] Paley R.E.A.C., Quar. J. Math. 4 pp 52– (1933) · Zbl 0006.24703 [16] Stepanov S.A., Mat. Zametki 32 pp 753– (1982) [17] Vaserstein L., J. Number Theory 26 pp 286– (1987) · Zbl 0624.10049 [18] Vaserstein, L. Ramsey’s theorem and Waring’s problem for algebras over fields. Proceedings of the Workshop at The Ohio State University. June17–261991. pp.435–441. Berlin, New York: Walter de Gruyter. [19] Voloch J.F., Bol. Soc. Brasil. Mat. 16 pp 29– (1985) · Zbl 0612.10011 [20] Voloch J.F., J. Number Theory 73 pp 195– (1998) · Zbl 0916.11019 [21] DOI: 10.1006/jnth.1996.0071 · Zbl 0866.11066 [22] Wooley T.D., Ann. Math. 135 pp 131– (1992) · Zbl 0754.11026 [23] Zannier U., Acta Arith. 64 pp 87– (1993) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.