# The minimal number of roots of surface mappings and quadratic equations in free groups

@article{Bogatyi2001TheMN,
title={The minimal number of roots of surface mappings and quadratic equations in free groups},
author={Semeon Antonovich Bogatyi and Daciberg Lima Gonçalves and Heiner Zieschang},
journal={Mathematische Zeitschrift},
year={2001},
volume={236},
pages={419-452}
}
Abstract. Let $f \colon S_h \to S_g$ be a continuous mapping between orientable closed surfaces of genus h and g and let c denote the constant map $c \colon S_h \to S_g$ with $c(S_h) = c\in S_g$. Let $\varrho(f)$ be the minimal number of roots of f' among all maps f' homotopic to f, i.e. $\varrho(f) = \min \{|f'^{-1}(c)| : f' \simeq f \colon S_h \to S_g \}$. We prove that $\varrho(f) = \max \{\ell(f), d - (d\cdot\chi(S_g) - \chi(S_h) ) \}$ where \$\ell(f) =[ \pi_1(S_g) : f_{\#}(\pi_1(S_h… CONTINUE READING

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