# 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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