prof-joe/miura

miura's Issues

Miura Package for Divisor Class Group Arithmetic

pR=(kk,v,w)->kk[v,MonomialOrder=>{Weights => w, Weights=>toList(1 ..#w) }];

qR=(R,p)->R/ideal p

inv=L-> quotient(ideal first first entries gens gb L, L)

reduced=L->inv inv L

add=(J,K)-> reduced (J*K)

double=J->add(J,J)

multi=(J,m)->(
if m==0 then return ideal 1_I
else if m%2==0 then double multi(J,m//2)
else add(double multi(J,(m-1)//2),J)
)

--The Macaulay2 Package Miura contains functions that compute divisor class group arithmetic for nonsingular curves. The package reduces computation in a divisor class group to that in the ideal class group via the isomorphism. The underlying quotient ring should be over the ideal given by a nonsingular curve in the form of Miura. Although computing two integral ideals is not hard, we need to obtain an ideal such that the shortest Gr"{o}bner basis component is minimum in order to obtain the representative of the ideal class. Although the basic procedure is due to Arita, the source code has become much shorter using MaCaulay2. The package is useful not just for computation itself but also for understanding the divisor class group arithmetic from the ideal point of view.