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Timestamp:
Dec 18, 2013, 6:42:50 PM (9 years ago)
Branches:
Children:
1129351bca33bc8786735f0f6b3ef6e938ff6352
Parents:
Message:
`fix: format, sum->lsum in ellipticcovers.lib`
Location:
Singular/LIB
Files:
2 edited

Unmodified
Removed
• ## Singular/LIB/ellipticcovers.lib

• Property mode changed from `100755` to `100644`
 rb5bb857 We implement a formula for computing the number of covers of elliptic curves. It has beed obtained by proving mirror symmetry for arbitrary genus by tropical methods in [BBM]. A Feynman graph of genus g is a trivalent, connected graph of genus g (with 2g-2 vertices and 3g-3 edges). The branch type b=(b_1,...,b_(3g-3)) of a stable map is the for arbitrary genus by tropical methods in [BBM]. A Feynman graph of genus g is a trivalent, connected graph of genus g (with 2g-2 vertices and 3g-3 edges). The branch type b=(b_1,...,b_(3g-3)) of a stable map is the multiplicity of the the edge i over a fixed base point. Given a Feynman graph G and a branch type b, we obtain the number Given a Feynman graph G and a branch type b, we obtain the number N_(G,b) of stable maps of branch type b from a genus g curve of topological type G to the elliptic curve by computing a path integral to the elliptic curve by computing a path integral over a rational function. The path integral is computed as a residue. KEYWORDS: tropical geometry; mirror symmetry; tropical mirror symmetry; Gromov-Witten invariants; elliptic curves; propagator; Feynman graph; path integral computeConstant(number, number)           constant coefficient in the Laurent series expansion of a rational function in a given variable evalutateIntegral(number, list)           path integral for a given propagator and ordered sequence of variables gromovWitten(number)                      sum of path integrals for a given propagator over all orderings of the variables, or gromovWitten(number)                      sum of path integrals for a given propagator over all orderings of the variables, or Gromov Witten invariant for a given graph and a fixed branch type, or list of Gromov Witten invariants for a given graph and all branch types partitions(int, int)                      partitions of an integer into a fixed number of summands permute(list)                             all permutations of a list sum(list)                                 sum of the elements of a list lsum(list)                                sum of the elements of a list "; product of propagator(list(v[i],w[i]),b[i]) over all edges i with multiplicity b[i] over the base point and vertices v[i] and w[i]. KEYWORDS: propagator; elliptic curve KEYWORDS: elliptic curve EXAMPLE:  example propagator; shows an example " { "EXAMPLE:"; echo=2; ring R=(0,x1,x2,x3,x4),(q1,q2,q3,q4,q5,q6),dp; graph G = makeGraph(list(1,2,3,4),list(list(1,3),list(1,2),list(1,2),list(2,4),list(3,4),list(3,4))); graph G = makeGraph(list(1,2,3,4),list(list(1,3),list(1,2),list(1,2),list(2,4),list(3,4),list(3,4))); propagator(list(x1,x2),0); propagator(list(x1,x2),2); RETURN:  number, the constant coefficient of the Laurent series of f in the variable x. THEORY:  Computes the constant coefficient of the Laurent series by iterative differentiation. KEYWORDS: Laurent series EXAMPLE:  example computeConstant; shows an example { "EXAMPLE:"; echo=2; ring R=(0,x1,x2,x3,x4),(q1,q2,q3,q4,q5,q6),dp; graph G = makeGraph(list(1,2,3,4),list(list(1,3),list(1,2),list(1,2),list(2,4),list(3,4),list(3,4))); graph G = makeGraph(list(1,2,3,4),list(list(1,3),list(1,2),list(1,2),list(2,4),list(3,4),list(3,4))); number P = propagator(G,list(1,1,1,0,0,0)); computeConstant(P,x2); In the setting of covers of elliptic curves this is the path integral over the propagator divided by the product of all variables (corresponding to the vertices) propagator divided by the product of all variables (corresponding to the vertices) computed as a residue. KEYWORDS: residue; Laurent series EXAMPLE:  example evaluateIntegral; shows an example { "EXAMPLE:"; echo=2; ring R=(0,x1,x2,x3,x4),(q1,q2,q3,q4,q5,q6),dp; graph G = makeGraph(list(1,2,3,4),list(list(1,3),list(1,2),list(1,2),list(2,4),list(3,4),list(3,4))); graph G = makeGraph(list(1,2,3,4),list(list(1,3),list(1,2),list(1,2),list(2,4),list(3,4),list(3,4))); number p = propagator(G,list(0,2,1,0,0,1)); evaluateIntegral(p,list(x1,x3,x4,x2)); This will eventually be deleted and become a more efficient kernel function. KEYWORDS: permutations EXAMPLE:  example permute; shows an example This may eventually be deleted and become a more efficient kernel function. KEYWORDS: partitions EXAMPLE:  example partitions; shows an example - the invariant N_(G,d)*|Aut(G)| where d is the degree of the covering, or @* - the number N_(G,b) of coverings with source G and target an elliptic curves with branch type a over a - the number N_(G,b) of coverings with source G and target an elliptic curves with branch type a over a fixed base point (that is, the i-th edge passes over the base point with multiplicity b[i]).@* KEYWORDS: Gromov-Witten invariants; elliptic curves; coverings; Hurwitz numbers EXAMPLE:  example gromovWitten; shows an example //print(string(j)+" / "+string(size(pa))+"    "+string(pa[j])+"     "+string(re[j])+"      "+string(sum(re))+"     "+string(ti)); } return(sum(re)); return(lsum(re)); } } { "EXAMPLE:"; echo=2; ring R=(0,x1,x2,x3,x4),(q1,q2,q3,q4,q5,q6),dp; graph G = makeGraph(list(1,2,3,4),list(list(1,3),list(1,2),list(1,2),list(2,4),list(3,4),list(3,4))); graph G = makeGraph(list(1,2,3,4),list(list(1,3),list(1,2),list(1,2),list(2,4),list(3,4),list(3,4))); number P = propagator(G,list(0,2,1,0,0,1)); gromovWitten(P); "USAGE:  computeGromovWitten(G, d, st, en [, vb] ); G graph, d int, st int, en  int, optional: vb int@* ASSUME:  G is a Feynman graph, d a non-negative integer, st specified the start- and en the end partition in the list pa = partition(d). Specifying a positive optional integer vb leads to intermediate printout.@* in the list pa = partition(d). Specifying a positive optional integer vb leads to intermediate printout.@* We assume that the coefficient ring has one rational variable for each vertex of G.@* RETURN:  list L, where L[i] is gromovWitten(G,pa[i]) and all others are zero. THEORY:  This function does essentially the same as the function gromovWitten, but is designed for handling complicated examples. Eventually it will also run in parallel.@* KEYWORDS: Gromov-Witten invariants; elliptic curves; coverings; Hurwitz numbers EXAMPLE:  example computeGromovWitten; shows an example re[j]=gromovWitten(propagator(P,pa[j])); ti=timer-ti; if (vb>0){print(string(j)+" / "+string(size(pa))+"    "+string(pa[j])+"     "+string(re[j])+"      "+string(sum(re))+"     "+string(ti));} if (vb>0){print(string(j)+" / "+string(size(pa))+"    "+string(pa[j])+"     "+string(re[j])+"      "+string(lsum(re))+"     "+string(ti));} } else {re[j]=s;} } ring R=(0,x1,x2,x3,x4),(q1,q2,q3,q4,q5,q6),dp; graph G = makeGraph(list(1,2,3,4),list(list(1,3),list(1,2),list(1,2),list(2,4),list(3,4),list(3,4))); partitions(6,2); partitions(6,2); computeGromovWitten(G,2,3,7); computeGromovWitten(G,2,3,7,1); proc sum(list L) "USAGE:  sum(L); L list@* ASSUME:  L is a list of things with the binary operator + defined.@* proc lsum(list L) "USAGE:  lsum(L); L list@* ASSUME:  L is a list of things with the binary operator + defined.@* RETURN:  The sum of the elements of L. THEORY:  Sums the elements of a list. THEORY:  Sums the elements of a list. Eventually this will be deleted and become a more efficient kernel function.@* KEYWORDS: sum EXAMPLE:  example sum; shows an example EXAMPLE:  example lsum; shows an example " { { "EXAMPLE:"; echo=2; list L = 1,2,3,4,5; sum(L); lsum(L); } "USAGE:  generatingFunction(G, d); G graph, d int@* ASSUME:  G is a Feynman graph, d a non-negative integer. The basering has one polynomial variable for each edge, and the coefficient ring has one rational variable for each vertex.@* edge, and the coefficient ring has one rational variable for each vertex.@* RETURN:  poly. THEORY:  This function compute the multivariate generating function of all Gromov-Witten invariants up to degree d, that is, the sum of all gromovWitten(G,b)*q^b.@* KEYWORDS: generating function; Gromov-Witten invariants; elliptic curves; coverings; Hurwitz numbers EXAMPLE:  example generatingFunction; shows an example
• ## Singular/LIB/primdecint.lib

 rb5bb857 //=== this is needed because quotient(I,f) does not work properly, should be //=== replaced by quotient later if ( f==0 ) { return( ideal(1) ); } if ( f==0 ) { return( ideal(1) ); } def R=basering; int i; ideal I=quotient(N,freemodule(nrows(N))); if(size(I)==0){return(list(list(N,I)));} list B=minAssZ(I); list S,R,L; else { // this is the case that P=

, p prime // this is the case that P=

, p prime I=std(I); ideal IC=simplify(flatten(lead(I)),2);

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