Editing Algebraic function (section)

=== Monodromy ===
Note that the foregoing proof of analyticity derived an expression for a system of ''n'' different '''function elements''' ''f''<sub>''i''{{space|hair}}</sub>(''x''), provided that ''x'' is not a '''critical point''' of ''p''(''x'',&thinsp;''y'').  A ''critical point'' is a point where the number of distinct zeros is smaller than the degree of ''p'', and this occurs only where the highest degree term of ''p'' or the [[discriminant]] vanish.  Hence there are only finitely many such points ''c''<sub>1</sub>,&thinsp;...,&thinsp;''c''<sub>''m''</sub>.

A close analysis of the properties of the function elements ''f''<sub>''i''</sub> near the critical points can be used to show that the  [[monodromy theorem|monodromy cover]] is [[Ramification (mathematics)|ramified]] over the critical points (and possibly the [[Riemann sphere|point at infinity]]).  Thus the [[Holomorphic function|holomorphic]] extension of the ''f''<sub>''i''</sub> has at worst algebraic poles and ordinary algebraic branchings over the critical points.

Note that, away from the critical points, we have

:<math>p(x,y) = a_n(x)(y-f_1(x))(y-f_2(x))\cdots(y-f_n(x))</math>

since the ''f''<sub>''i''</sub> are by definition the distinct zeros of ''p''.  The [[monodromy group]] acts by permuting the factors, and thus forms the '''monodromy representation''' of the [[Galois group]] of ''p''.  (The [[monodromy action]] on the [[universal covering space]] is related but different notion in the theory of [[Riemann surface]]s.)