Abstract:
It is considered such geometric invariants of an algebraic curve as the minimal number of crucial values of the rational functions and the minimal transcendence degree of the definition fields. The question is if the difference of these two invariants is always equal to 3 for any curve with the genus $g>0$. For curves defined over an algebraic number field the positive answer is given by Belyi's theorem. In the paper the positive answer is given for some other cases.
Key words and phrases:algebraic curve, definition field, Belyi theorem, projective line, transcendence degree, ramification, critical value,
rational function.
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