Geodesic
Minimal Number of Preimages Under Maps of Surfaces
Matematičeskie zametki, Tome 75 (2004) no. 1, pp. 13-19.

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We compute the minimal number of roots of the equation $f(x)=c$ for a map $f(x)$ belonging to a given homotopy class of maps between closed (not necessarily oriented) surfaces. 
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     title = {Minimal {Number} of {Preimages} {Under} {Maps} of {Surfaces}},
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S. A. Bogatyi; D. L. Gonçalves; E. A. Kudryavtseva; H. Zieschang. Minimal Number of Preimages Under Maps of Surfaces. Matematičeskie zametki, Tome 75 (2004) no. 1, pp. 13-19. http://geodesic.mathdoc.fr/item/MZM_2004_75_1_a1/

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