Given a general Equation of a Reimann surface of the form:
F(X, Y)= (X10n*Y14n+X14n*Y10n)+(X9n + Y9n) +(X8n +Y8n) -1;
The Genus "g" for the above Riemann Surface of some Automorphism Group, have been computed by a program called MAGMA from University of Sydney in Australia for all values of: n =1, 2, ....12.
And the results as follows:
n = 1, g(1 )= 115
n = 2, g(2) = 489
n = 3, g(3) = 1123
n = 4, g(4) = 2017
n = 5, g(5) = 3171
n = 6, g(6) = 4585
n = 7, g(7) = 6259
n = 8, g(8) = 8193
n = 9, g(9) = 10387
n = 10, g(10) = 12926
n = 11, g(11) = 15555
n = 12, g(12) = 18529
The question for all is how to find a formula for the genus g(n) in terms of n, which satisfies all the above computed values and can be used to find any value of the genus for all n > 12.
Dear Reza Zomorrodian , Determine the degree of the curve, use the Genus Formula for smooth plane curves. Consider smoothness. Assume smoothness 288n2−36n+1, check whether the curves are smooth. KR Rob
The attachment has the function and this polynomial exactly fits all the given values for n = 1 to n = 12 and can be used to compute genus g(n) for any n > 12 as well.
Dear Reza Zomorrodian ,
To find the genus of a very special Riemann surface, you often need to leverage its geometric and algebraic properties, especially those related to its symmetries (automorphisms) and the curves that define it. For example, if the Riemann surface is defined by a polynomial equation, you might be able to use the degree of the polynomial or properties of its singularities to determine the genus.
Chapter Book Reviews: Riemann Surfaces of Infinite Genus by J. Feldm...
Regards,
Shafagat
Dear Jayaseelan Gunaseelan: That's a great possible formula I was looking for. There is only one thing. I needed the g(13) and for n=13, my MAGAM program took six days to compute the value that I was looking for in my up coming paper. The program MAGMA gave: g(13)= 21763, now I am afraid of testing your fund formula to see If I get the same answer g(13)=21763 ? Would you please test your formula for my Riemann surface of genus g(13)= 21763? Then you have given the perfect answer.
Hello again all:
The obtain formula must satisfy my discovered Riemann Surface of genus g(13)= 21763. So if any of you found a formula for my paper, be sure to tested to see if the genus of the surface for n=13, actually 21763.
Thank you all!!
Dear Reza Zomorrodian , the best approximation is cubic fit : g(n)≈−0.2312n3+134.5n2−38.58n+26.76. The genus values likely come from a quotient surface under an automorphism group. I hope this is of help for you. KR Rob
I checked the formula obtained by Jayaseelan Gunaseelan, and it does work only for the first 12 order pairs. But it does not work for my important genus of g(13) = 21763.
The formula obtained by Jayaseelan Gunaseelan, gives:
g(13) = 40463 and not the genus I am looking for which must be 21763.
What is the reason for that. MAGMA gives the genus I was looking for, i.e. 21763 for n= 13
Doing the group G(5, 31):-
Finitely presented group G on 2 generators
Relations
G.1^2 = Id(G)
G.2^3 = Id(G)
(G.1 * G.2)^78 = Id(G)
(G.1, G.2)^93 = Id(G)
(G.2^-1, G.1)^3 = (G.1, G.2)^15
Time: 1.990
|G|= 282906= 13( 21763 - 1) which means the genus g = 21763
I found the Riemann surface of genus 21763, oddly enough after long long time search to be:
(X130*Y182 + X182*Y130) + (X117 + Y117) + (X104 + Y104)=1