View source for Chasles-Cayley-Brill formula

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c1300701.png" /> be an irreducible algebraic plane curve of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c1300702.png" />, given by an equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c1300703.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c1300704.png" /> is an irreducible bivariate polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c1300705.png" /> over a ground field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c1300706.png" /> (cf. also [[Algebraic curve|Algebraic curve]]). For simplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c1300707.png" /> is assumed to be algebraically closed (cf. also [[Algebraically closed field|Algebraically closed field]]), although most of what is said below can be suitably generalized without that assumption. For the basic field theory involved, see [[#References|[a6]]] (or the modernized version [[#References|[a4]]]) and [[#References|[a3]]]. For much of the geometry to be discussed, see [[#References|[a5]]] and [[#References|[a8]]]; in particular, for the idea of points at infinity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c1300708.png" />, see [[#References|[a1]]]. For an interplay between the geometry and the algebra, see [[#References|[a2]]].

One starts by analyzing when the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c1300709.png" /> can be rationally parametrized. For example, the unit circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007010.png" /> has the rational parametrization

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Likewise, the cuspidal cubic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007012.png" /> has the rational parametrization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007014.png" />. However, the non-singular cubic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007015.png" /> does not have any rational parametrization. To obtain the parametrization of the circle, one cuts it by a line of slope <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007016.png" /> through the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007017.png" /> and notes that it meets the circle in the variable point

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For the cuspidal cubic one takes a line through the cusp <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007019.png" /> and notes that it meets the cubic in the variable point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007020.png" />. This works because a line meets a circle in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007021.png" /> points, and it meets a cubic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007022.png" /> points. In case of a cuspidal cubic two intersections are absorbed in the cusp. In case of a non-singular cubic there is no such point for the absorption. Generalizing this one can show that the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007023.png" /> cannot have more than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007024.png" /> double points and if it does have that many, then it can be parametrized rationally. To this end one first notes that a bivariate polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007025.png" /> has

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coefficients and hence the dimension of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007027.png" /> of curves of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007028.png" /> passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007029.png" /> double points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007030.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007031.png" />. Next, by the [[Bezout theorem|Bezout theorem]] (which is the oldest theorem in [[Algebraic geometry|algebraic geometry]]), a curve of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007032.png" /> and a curve of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007033.png" />, having no common component, meet in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007034.png" /> points, counted properly. In the proper counting a double point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007035.png" /> should be counted twice. Thus, the number of free points in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007036.png" /> meets a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007037.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007038.png" /> passing through the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007039.png" /> double points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007040.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007041.png" />. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007042.png" /> is also the dimension of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007043.png" />, the members of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007044.png" /> which pass through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007045.png" /> fixed simple points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007046.png" /> form a pencil, i.e., a one-parameter family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007047.png" />, a variable member of which meets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007048.png" /> in one variable point whose coordinates are single-valued, and hence rational, functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007049.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007050.png" /> had an extra double point, then one can take a value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007051.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007052.png" /> so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007053.png" /> goes through it and this would make the properly counted intersections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007055.png" /> to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007056.png" />, contradicting the Bezout theorem because <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007057.png" /> is irreducible and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007058.png" /> has smaller degree.

Provisionally defining the genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007059.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007060.png" /> (cf. also [[Genus of a curve|Genus of a curve]]) by

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one always has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007062.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007063.png" /> is rational, i.e., has a rational parametrization. To make the reverse implication <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007064.png" /> also true, one must learn to count the double points properly. To begin with, one must include singularities of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007065.png" /> at infinity. Next, by looking at the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007066.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007067.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007068.png" />, which is obviously rational and has a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007069.png" />-fold point at the origin and an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007070.png" />-fold point at infinity as its only singularities, one decides to count a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007071.png" />-fold point as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007072.png" /> double points. Before discussing infinitely near singularities, one notes that the degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007073.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007074.png" /> can be geometrically characterized as the number of points in which a general line meets it.

Likewise, the multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007075.png" /> of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007076.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007077.png" /> can be characterized geometrically by saying that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007078.png" /> is equal to the number of points in which a generic line through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007079.png" /> meets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007080.png" /> outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007081.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007082.png" /> is a simple or singular point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007083.png" /> according as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007084.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007085.png" />. Algebraically, by translating the coordinates one may assume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007086.png" /> to be the origin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007087.png" />, and then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007088.png" /> is the order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007089.png" />, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007090.png" /> has terms of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007091.png" /> but none of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007092.png" />. By making the quadratic transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007093.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007094.png" /> one gets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007095.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007096.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007097.png" /> is the proper transform of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007098.png" />. The exceptional line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c13007099.png" /> meets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070100.png" /> in points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070101.png" /> whose multiplicities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070102.png" /> add up to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070103.png" />. These are the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070104.png" /> in the first neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070105.png" />. Points in the first neighbourhoods of these points are the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070106.png" /> in the second neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070107.png" />, and so on. It is easily seen that all points in a high enough neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070108.png" /> are simple. Now <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070109.png" /> is counted as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070110.png" /> double points, where

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with the summation extended over all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070112.png" /> in the various neighbourhoods of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070113.png" />, including <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070114.png" />; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070115.png" /> is the multiplicity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070116.png" />; clearly: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070117.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070118.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070119.png" /> is a simple point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070120.png" />. One arrives at the exact genus formula

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with summation over all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070122.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070123.png" />. One always has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070124.png" />; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070125.png" /> is rational.

It turns out that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070126.png" /> is a birational invariant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070127.png" />, i.e., it remains unchanged when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070128.png" /> undergoes a birational transformation (cf. also [[Birational morphism|Birational morphism]]). The residue class ring of the polynomial ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070129.png" /> modulo the ideal generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070130.png" /> is the affine coordinate ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070131.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070132.png" />. Note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070133.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070134.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070135.png" /> are the images of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070136.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070137.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070138.png" />. The quotient field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070139.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070140.png" /> is the function field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070141.png" />. A birational correspondence between curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070142.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070143.png" /> is an almost one-to-one correspondence; it is given by a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070144.png" />-isomorphism between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070145.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070146.png" />. So one should be able to define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070147.png" /> directly in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070148.png" />. Following C.G.J. Jacobi one takes any differential of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070149.png" /> (cf. also [[Differential field|Differential field]]), i.e., an expression of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070150.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070151.png" />, and shows that if the differential is not zero, then the number of its zeros minus the number of its poles equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070152.png" />. Having brought the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070153.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070154.png" /> to the origin, its local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070155.png" /> is defined to be the subring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070156.png" /> consisting of all quotients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070157.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070158.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070159.png" /> are polynomials with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070160.png" /> (cf. also [[Local ring|Local ring]]); its unique [[Maximal ideal|maximal ideal]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070161.png" /> consists of the above quotients with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070162.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070163.png" /> be the conductor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070164.png" />, i.e., the largest ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070165.png" /> which remains an ideal in the integral closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070166.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070167.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070168.png" />. It can be shown that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070169.png" /> is the length of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070170.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070171.png" />, i.e., the maximal length of strictly increasing chains of ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070172.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070173.png" />; moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070174.png" /> is the length of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070175.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070176.png" />, which is a ubiquitous result having two dozen proofs in the literature. The ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070177.png" /> has a finite number of maximal ideals and localizing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070178.png" /> at them gives discrete valuation rings; as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070179.png" /> varies over all points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070180.png" />, including those at infinity, these discrete valuation rings vary over the [[Riemann surface|Riemann surface]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070181.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070182.png" />, i.e., the set of all discrete valuation rings whose quotient field is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070183.png" /> and which contain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070184.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070185.png" /> denote the localizations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070186.png" /> at the various maximal ideals in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070187.png" /> (cf. also [[Localization in a commutative algebra|Localization in a commutative algebra]]); one calls <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070188.png" /> the centre on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070189.png" /> of the members of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070190.png" />; note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070191.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070192.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070193.png" /> is a simple point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070194.png" />, and hence for all except a finite number of points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070195.png" />, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070196.png" /> has exactly one member. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070197.png" /> and non-zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070198.png" /> one puts

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070199.png" /></td> </tr></table>

with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070200.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070201.png" />; take <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070202.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070203.png" /> and define

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070204.png" /></td> </tr></table>

one calls <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070205.png" /> a uniformizing parameter of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070206.png" />. Now the number of zeros minus number of poles of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070207.png" /> equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070208.png" /> taken over all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070209.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070210.png" />. For any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070211.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070212.png" />, not at infinity, one has Dedekind's formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070213.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070214.png" /> is the different ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070215.png" /> defined by saying that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070216.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070217.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070218.png" />.

For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070219.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070220.png" /> be an irreducible algebraic plane curve such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070221.png" /> is a finite separable algebraic field extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070222.png" /> of field degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070223.png" /> (cf. also [[Extension of a field|Extension of a field]]; [[Separable extension|Separable extension]]). This defines a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070224.png" /> correspondence between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070225.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070226.png" />, and hence between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070227.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070228.png" />; namely, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070229.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070230.png" /> correspond if and only if for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070231.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070232.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070233.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070234.png" /> be the genus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070235.png" />, let the different <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070236.png" /> be the integer-valued function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070237.png" /> whose value at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070238.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070239.png" /> is given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070240.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070241.png" /> is a uniformizing parameter of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070242.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070243.png" /> with summation over all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070244.png" />. Then the Riemann–Hurwitz formula says that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070245.png" /></td> </tr></table>

and this gives rise to the Zeuthen formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070246.png" /></td> </tr></table>

Now suppose there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070247.png" />-isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070248.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070249.png" /> is called a fixed place of the correspondence if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070250.png" />. The Chasles–Cayley–Brill formula says that under suitable conditions, the number of these, counted properly, equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070251.png" />, where the integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070252.png" /> is called the valence of the correspondence. For details see [[#References|[a7]]], pp. 189–194.

In case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070253.png" /> is the field of complex numbers, to describe Riemann's approach one topologizes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070254.png" /> to make it into a compact orientable two-dimensional real manifold, and hence into a sphere with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070255.png" /> handles (cf. also [[Riemann surface|Riemann surface]]). Likewise, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070256.png" /> is made into a sphere with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070257.png" /> handles. Triangulate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070258.png" /> by including all the branch points as vertices, and lift this triangulation to a triangulation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070259.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070260.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070261.png" /> be the vertices, edges, faces of the bottom and top triangulations respectively. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070262.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070263.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070264.png" />, and hence by the Euler–Poincaré theorem one obtains

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070265.png" /></td> </tr></table>

This proves the birational invariance of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130070/c130070266.png" /> and the Riemann–Hurwitz formula. For details, see [[#References|[a2]]] and [[#References|[a4]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S.S. Abhyankar,   "What is the difference between a parabola and a hyperbola"  ''Math. Intelligencer'' , '''10'''  (1988)  pp. 36–43</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S.S. Abhyankar,   "Algebraic geometry for scientists and engineers" , Amer. Math. Soc.  (1990)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S.S. Abhyankar,   "Field extensions"  G.A. Pilz (ed.)  A.V. Mikhalev (ed.) , ''Handbook of the Heart of Algebra'' , Kluwer Acad. Publ.  (to appear)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  C. Chevalley,   "Introduction to the theory of algebraic functions of one variable" , ''Math. Surveys'' , '''6''' , Amer. Math. Soc.  (1951)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J.L. Coolidge,   "A treatise on algebraic plane curves" , Clarendon Press  (1931)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  R. Dedekind,   H. Weber,   "Theorie der algebraischen Functionen einer Veränderlichen"  ''Crelle J.'' , '''92'''  (1882)  pp. 181–290</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  S. Lefschetz,   "Algebraic geometry" , Princeton Univ. Press  (1953)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  F. Severi,   "Vorlesungen über algebraische Geometrie" , Teubner  (1921)</TD></TR></table>