|
 | ||
| ||
| ||
| ||
|
| ||
| ||
| ||
| ||
|
| ||
| ||
|
Contents of this page
General information on the library
jinvariant.lib is a SINGULAR library for computations needed in the proof of the main results in
Eric Katz, Hannah Markwig, Thomas Markwig: The j-Invariant of a Plane Tropical Cubic.
Preprint 2007,
and in
which is the following:
If (Q,S) is a marked polygon with one interior point, then a general polynomial f in K[x,y] with support S defines an elliptic curve C on the toric surface XS. If K has a non-archimedean valuation into the reals we can tropicalize C to get a tropical curve Trop(C). If the Newton subdivision induced by f is a triangulation, then Trop(C) will be a graph of genus one and we show that the lattice length of the cycle of that graph is the negative of the valuation of the j-invariant of C.
For the definitions and the details we refer to the above papers.The library can be downloaded from
If you use the library in your scientific work and want to cite it, please do so as follows:
Eric Katz,
Hannah Markwig,
Thomas Markwig:
jinvariant.lib. A
SINGULAR
3.0 library for computations with
j-invariants in tropical geometry, 2007
If you encounter any problems using the library please send an email to
On this web page you will find some information on the procedures in the library jinvariant.lib and the post script files produced by them. Nearly no knowledge of SINGULAR is necessary in order to use the procedures of the library.
The main results obtained by the procedures in this library are contained in the post script files:
| • | discriminant_fan_of_cubic.ps (produced by displayfan) |
| • | secondary_fan_of_cubic.ps (produced by displayfan) |
| • | raysC.ps (produced by raysC) |
| • | discriminant_fan_of_2x2.ps (produced by displayfan) |
| • | secondary_fan_of_2x2.ps (produced by displayfan) |
| • | discriminant_fan_of_4x2.ps (produced by displayfan) |
| • | secondary_fan_of_4x2.ps (produced by displayfan) |
Requirements
In order to use the library jinvariant.lib you need to have the libraries tropical.lib and polymake.lib as well which can be obtained via respectively via This library contains a number of procedures which might be useful when dealing with j-invariants for tropical curves, such as the procedure tropicalJInvariant which actually computes the cycle length of a tropical curve with cycle, or the procedure drawTropicalCurve which draws a tropical curve given by a polynomial over Q(t).
Even though this is a SINGULAR library, many of the hard computations involving polytopes are done by calling two other programs, namely
| • | polymake by Ewgenij Gawrilow, TU Berlin and Michael Joswig, TU Darmstadt see http://www.math.tu-berlin.de/polymake, and |
| • | topcom by Joerg Rambau, Universitaet Bayreuth see http://www.uni-bayreuth.de/departments/wirtschaftsmathematik/rambau/TOPCOM. |
Whenever one of these programs is used, we assume that there is a directory /tmp which is accessible for the user to write and read data. The output produced by polymake respectively topcom can then be found in the directory /tmp and are available for further computations.
Moreover, some procedures produce latex files and compile them to display post script output. Again we assume the presence of /tmp, and we assume that the programs latex, dvips and kghostview are present as well.
How to use SINGULAR ?
First we want to give some general advise for users not familiar with SINGULAR , for more information see SINGULAR .
| • | To start Singular, type the command Singular in a shell. You must do this in the directory where your copies of the libraries tropical.lib, polymake.lib and jinvariant.lib reside. The program will start and give you some information on the version and the authors. Moreover, it will produce a prompt waiting for your commands. | ||||||||
| • |
To load the libraries type
LIB "jinvariant.lib";
at the prompt -- this will automatically also load tropical.lib
and polymake.lib. Note, every command in Singular
ends with a semicolon.
|
||||||||
| • |
Most of the procedures in the library need one or two strings as
input, some need no input at all, and some need integers.
It is thus not necessary
to know the concept of a ring in
SINGULAR
which usually is vital for all
computations with
SINGULAR
. To call a procedure which requires no
input, e.g. raysC, just type the name of the procedure
followed by the brackets () and the semicolon at the prompt, e.g.
raysC();
To call a procedure which takes two strings as input, e.g. fan,
type the name of the procedure followed by the input in brackets
followed by semicolon and note that a string has always starting and
ending quotes, e.g
fan("cubic","discriminant");
To call a procedure which takes integers as input, e.g. examplesC,
type the name of the procedure followed by the input in brackets
followed by semicolon, e.g
examplesC(5);
|
||||||||
| • |
Every object in
SINGULAR
has its own type and this type has to be specified when defining the
object. In particular, the return value of a procedure has a type, and
if you want to store the return value then you have to assign the
result to a variable of the corresponding type, e.g
list FAN=fan("cubic","discriminant",1);
creates a variable of type list and assigns to it the output of
the procedure inequalititesC.
The types needed to work with this library are list, which is an array of other objects, int defining an integer, intvec defining a vector of integers, and intmat defining an integer matrix. Note that there are procedures which have no return value. |
||||||||
| • |
|
||||||||
| • |
You can invoke the
SINGULAR
help on the library by the command
help jinvariant.lib;
This will give you a list of all procedures available in the library with
a very short description on what they are supposed to do. If you want information on
a particular procedure, e.g. on fan, type
help fan;
The help will be displayed in the shell window which you are running. Instead you can
open the library in a text editor and search for proc fan. There you
will find the same help string at the very beginning of the library. If you need help
on general
SINGULAR
commands or features (e.g. on intmat), just type
help intmat;
The help should then in general be displayed in a web browser. If this does not work for
some reason, then simply go the
SINGULAR
web page and look in the online manual.
|
||||||||
Information on the considered elliptic curves
The proof of the result mentioned above for all marked polygons with exactly one interior point is reduced to the study of three cases, namely cubics and curves of type 2x2 respectively of type 4x2. These are given by either a cubic polynomial
f=u30x3+u21x2y+u12xy2+u03y3+u20x2+u11xy+u02y2+u10x+u01y+u00,
or a type 2x2 polynomial
f=u22x2y2+u21x2y+u12xy2+u20x2+u11xy+u02y2+u10x+u01y+u00,
or a type 4x2 polynomial
f=u40x4+u30x3+u21x2y+u20x2+u11xy+u02y2+u10x+u01y+u00,
where the coefficients uij are in a field K with
non-archimedean valuation into the real numbers. Whenever we want to
compute examples we use as base field the quotient field Q(t) of the
polynomial ring Q[t] and consider it as a subfield of the field C{{t}} of
Puiseux series over the complex numbers. The field C{{t}} is in a
natural way equipped with a
valuation associating to a Puiseux series its order.
We extend the valuation on K by componentwise application to the torus (K*)2 in which the elliptic curve C given as the zero set of f lives, giving a map to R2. The topological closure of its image is by Kapranov's Theorem the tropical curve Trop(C) defined the tropical polynomial
Trop(f)=min{u30+3x,u21+2x+y,u12+x+2y,u03+3y,u20+2x,u11+x+y,u02+2y,u10+x,u01+y,u00},
respectively
Trop(f)=min{u22+2x+2y,u21+2x+y,u12+x+2y,u20+2x,u11+x+y,u02+2y,u10+x,u01+y,u00},
respectively
Trop(f)=min{u40+4x,u30+3x,u21+2x+y,u20+2x,u11+x+y,u02+2y,u10+x,u01+y,u00},
where by abuse of notation we denote her by uij the
valuations of the coefficients uij. This should not lead to
any confusion, and we will use this convention in the
procedures as well.
We are mainly studying the j-invariant of the elliptic curve defined by f. It can be expressed as a quotient of two homogeneous polynomials of degree 12 where the variables are just the coeffients of f, and again we use just the notation uij. We consider them ordered as follows:
u11,u30,u20,u10,u00,u21,u01,u12,u02,u03,
respectively
u11,u20,u10,u00,u21,u01,u22,u12,u02,
respectively
u11,u40,u30,u20,u10,u00,u21,u01,u02.
The denominator of the j-invariant is for cubics and type 2x2 curves
the discriminant of f and for type 4x2 curves it is
u022 times the discriminant of f. This denominator
is the object with which most of the procedures are concerned,
computing its Newton polytope and its Gröbner fan, and studying
certain cones in the latter which correspond to tropical polynomials
whose tropical curve has a cycle.
Among other things we show that all these cones lie in a single cone
of the Gröbner fan of the numerator of the j-invariant.
We also compute on each of these cones the generic valuation of the j-invariant and the cycle length of the tropical curve and compare them. This is done by the procedure displayfan and the resulting post script file is available via respectively respectively Alternatively we can consider the secondary fan of the set S of markings which is a refinement of the Gröbner fan of the denominator, and we have done so, since this lists for each cone of the Gröbner fan of the denominator all corresponding triangulations. We again compute on each of these cones the generic valuation of the j-invariant and the cycle length of the tropical curve and compare them. This is done by the procedure displayfan and the resulting post script file is available via respectively respectively Finally in the first of the above mentioned papers the rays of the secondary fan associated to a plane cubic are classified, and we compute for each of these rays the generalised cycle length as well as the generic valuation of the j-invariant. This is done by the procedure raysC and the resulting post script file is available via
The main procedures of the library
| displayFan | produces a post script file showing the information computed with 'fan' |
| fan | computes the Groebner fan of the discriminant or the secondary fan |
| polytope | computes Newton polytopes respectively secondary polytopes |
| testInteriorInequalities | compares a cone for the numerator of the j-invariant to another cone |
Detailed description of the main procedures of the library
1) displayFan(string,string)
| Warning: |
The procedure requires the program
polymake
to be installed on your computer.
Moreover, it assumes that the directory /tmp exists and that the programs latex, dvips and kghostview are available. |
|||||||||||||||
| Short: | The procedure produces a post script file showing the information computed with fan. | |||||||||||||||
| Long: | In order to prove the above mentioned statement that for elliptic curves over the Puiseux series the generic valuation of the j-invariant coincides with the negative of the cycle length of the tropical curve associated to that curve provided it has a cycle, we work through all cones of the Gröbner fan of the discriminant of the curve which correspond to tropical curves with cycle. We check the statement on each cone. The post script file produced by this procedure compares the two functions on each of the cones of interest and gives some information for the remaining cones as well. | |||||||||||||||
| Usage: | displayFan(polygon,type); | |||||||||||||||
| Assume: |
'polygon' is one of the following strings discribing the type of curve
to be considered:
|
|||||||||||||||
| Result: | The procedure has no return value. However, it produces the files /tmp/'filename'.tex and /tmp/'filename'.ps, and it displays the latter using kghostview, where 'filename' is the string 'polygon'_fan_of_'type' with 'polygon' and 'type' being the input data. |
2) fan(string,string)
| Warning: | The procedure requires the program polymake to be installed on your computer. Moreover, it assumes that the directory /tmp exists. | |||||||||||||||
| Short: | The input data refer either to the Gröbner fan of the discriminant of an elliptic curve defined by 'polygon' or the secondary fan of that polygon. The procedure computes the fan in question. | |||||||||||||||
| Usage: | fan(polygon,type); | |||||||||||||||
| Assume: |
'polygon' is one of the following strings discribing the type of curve
to be considered:
|
|||||||||||||||
| Result: |
The procedure returns a list, say L, whose first entry is again a list
such that
The second entry L[2] is an integer matrix describing the linearity space of the fan, i.e. its rows span a vector space which is contained in each cone of the fan. |
3) polytope(string,string)
| Warning: | The procedure requires the program polymake to be installed on your computer, and if the input 'type' described below is "secondary" then it needs the program topcom as well. Moreover, depending on your computer the computation might take more than a day in the latter case! | ||||||||||||||||||
| Short: | The input data refer either to the Newton polytope of the discriminant of an elliptic curve defined by 'polygon', or the the Newton polytope of the numerator of the j-invariant of that curve, or to the secondary polytope of that polygon. The procedure computes the polytope in question. | ||||||||||||||||||
| Usage: | polytope(polygon,type); | ||||||||||||||||||
| Assume: |
'polygon' is one of the following strings discribing the type of curve
to be considered:
|
||||||||||||||||||
| Result: |
The return value, say L, of the procedure is of type list with four
entries, namely:
|
||||||||||||||||||
| Remark: |
polygonToCoordinates( |
4) testInteriorInequalities(string)
| Warning: | The procedure requires the program polymake to be installed on your computer. | |||||||||
| Short: | The procedure checks if the cone U (the union of all cones of the secondary fan corresponding to marked subdivisions for which the interior point is visible) is completely contained in the one particular cone, say C, of the Gröbner fan of the numerator of the j-invariant of the curve specified by the input. | |||||||||
| Usage: | testInteriorInequalities(polygon); | |||||||||
| Assume: |
'polygon' is one of the following strings discribing the type of curve
to be considered:
|
|||||||||
| Result: | The return value of the procedure is of type list with three integer matricies as entries. The first integer matrix represents the inequalities of C, the second matrix contains the extreme rays of D, and the third integer matrix is their product. | |||||||||
| Remark: | The fact that the third matrix contains only non-negative entries shows that D is contained in C. |
Procedures only for plane cubics
| nonRefinementC() | shows that the secondary fan is no refinement for the numerator of the j-invariant |
| raysC() | compares cycle length and generic j-invariant on rays |
| examplesC(int) | computes examples of elliptic cubics |
Detailed description of the procedures only for plane cubics
5) nonRefinementC
| Warning: | The procedure requires the program polymake to be installed on your computer. |
| Long: | The procedure intersects the cone of the Groebner fan of the numerator of the j-invariant of a cubic which is dual to the vertex 12u_11 in the Newton polytope with the cone number 851 in the secondary fan of the cubic; it computes the dimension of the cone and its extreme rays; it then computes the same information just for the cone numer 851 in the secondary fan; it turns out that both cones have the same dimension but the extreme rays do not coincide (i.e. they intersect in a full dimensional cone which is strictly contained in the cone number 851 of the secondary fan); in particular, this proves that the secondary fan is not a refinement of the Gröbner fan of the numerator of the j-invariant of a cubic. |
| Usage: | The procedure is called without any argument, just type nonRefinementC(); at the prompt. |
| Result: |
The procedure returns a list of three integer matrices and two integers. The rows of
the first matrix are the extreme rays of the intersection of the above
mentioned cones, and the rows of the second matrix are the extreme
rays of the cone 851 in the secondary fan. The two integers are the
dimensions of these two cones.
The fact that the extreme rays do not conincide while the dimensions do, shows that the intersection is full dimensional but not the equal to the cone in the secondary fan. |
6) raysC
| Warning: |
The procedure requires the program
polymake
to be installed on your computer.
Moreover, it assumes that the directory /tmp exists and that the programs latex, dvips and kghostview are available. |
| Short: | The procedure produces the post script file raysC.ps. |
| Usage: | The procedure is called without any argument, just type raysC(); at the prompt. |
| Result: | The procedure has no return value. However, it produces the files /tmp/raysC.tex and /tmp/raysC.ps, and it displays the latter using kghostview. |
| Remark: | In the above mentioned paper among others the rays of the secondary fan associated to a cubic have been classified up to symmetry. For each such class of rays we have chosen one special representative and we compute the generalised cycle length and the generic valuation of its j-invariant. |
7) examplesC
| Short: | The procedure computes n random examples of plane cubic curves over the Puiseux series and it computes the valuation of the j-invariant as well as the cycle length of the corresponding tropical curve. |
| Usage: | The procedure is called with one integer n as only argument, e.g. examplesC(5);. |
| Result: | The procedure has no return value. |
| Remark: | For each example which it computes it produces files /tmp/example-f-'number'.tex, /tmp/example-wf-'number'.tex and /tmp/example-swf-'number'.tex, which show the tropicalisation of the example respectively of its Weierstrass form respectively of its reduced Weierstrass form. Moreover, it produces the file /tmp/examplesC.txt which contains the information on the cycle length and the valuation of the j-invariant for each of the examples as well as their equations. |
Procedures only for elliptic curves of type (2,2) on P1xP1
| examples2x2(int) | computes examples of elliptic curves of type (2,2) on P1xP1 |
Detailed description of the procedures for curves of type (2,2)
8) examples2x2
| Short: | The procedure computes n random examples of elliptic curves of type (2,2) over the Puiseux series and it computes the valuation of the j-invariant as well as the cycle length of the corresponding tropical curve. |
| Usage: | The procedure is called with one integer n as only argument, e.g. examples2x2(5);. |
| Result: | The procedure has no return value. |
| Remark: | For each example which it computes it produces files /tmp/example2x2-f-'number'.tex and /tmp/example2x2-wf-'number'.tex, which show the tropicalisation of the example respectively of its Weierstrass form. Moreover, it produces the file /tmp/examples2x2.txt which contains the information on the cycle length and the valuation of the j-invariant for each of the examples as well as their equations. |
Procedures accessing stored data
| polygonDB(string) | stores the points of the polygon |
| invariantsDB(string,string) | returns certain invariants of the type of elliptic curve specified by the input |
| affineHullDB(string,string) | stores the affine hull of the Groebner and secondary fans |
| verticesDB(string,string) | stores the vertices of the Groebner and secondary fans |
| vertexEdgeGraphDB(string,string) | stores the vertex edge graph of the Groebner and secondary fans |
| extremeRaysDB(int,string,string) | stores the extreme rays of the Groebner and secondary fans |
| triangulationsDB(int,string) | stores the triangulations of the polygons |
| edgeInequalitiesDB(string) | stores the inequalities for inner edges in a subdivision |
| interiorInequalitiesDB(string) | stores the inequalities for inner edges in a subdivision |
Detailed description of the procedures accessing stored data
9) polygonDB(string)
| Short: | The procedure returns the lattice points of the polygon in question, starting with the only interior point. | |||||||||
| Usage: | polygonDB(polygon); | |||||||||
| Assume: |
'polygon' is one of the following strings discribing the type of curve
to be considered:
|
|||||||||
| Result: | The return value of the procedure is a list of integer vectors, i.e. the lattice points of the polygon. |
10) invariantsDB(string,string)
| Short: | The procedure reproduces the discriminant, the invariant c4 respecively the numerator of the j-invariant of the type of elliptic curve specified by the input. | ||||||||||||||||||
| Usage: | invariantsDB(polygon,invar); | ||||||||||||||||||
| Assume: |
'polygon' is one of the following strings discribing the type of curve
to be considered:
|
||||||||||||||||||
| Result: | The return value of the procedure is a string, representing the invariant that was requested. | ||||||||||||||||||
| Remark: | The invariant is a polynomial in the unknowns uij, with i,j integers, where the uij can be thought of as coefficients of a defining affine equation. The invariants are not computed here, they are just stored. Note that j=c4^3/Delta=A/Delta is the j-invariant of the corresponding elliptic curve. |
11) affineHullDB(string,string)
| Short: | The procedure returns the affine hull of the Newton polytope of the discriminant respectively the secondary polytope. | |||||||||||||||
| Usage: | affineHullDB(polygon,type); | |||||||||||||||
| Assume: |
'polygon' is one of the following strings discribing the type of curve
to be considered:
|
|||||||||||||||
| Result: | The return value of the procedure is an integer matrix representing the affine hull of the polytope in question as a linear system of inequalities where the first column is the inhomogeneity. |
12) verticesDB(string,string)
| Short: | The procedure returns the vertices of the Newton polytope of the discriminant respectively the secondary polytope. | |||||||||||||||
| Usage: | verticesDB(polygon,type); | |||||||||||||||
| Assume: |
'polygon' is one of the following strings discribing the type of curve
to be considered:
|
|||||||||||||||
| Result: | The return value of the procedure is an integer matrix whose rows are the vertices of the polytope in question. |
13) vertexEdgeGraphDB(string,string)
| Short: | The procedure returns the vertex edge graph of the Newton polytope of the discriminant respectively the secondary polytope. | |||||||||||||||
| Usage: | vertexEdgeGraphDB(polygon,type); | |||||||||||||||
| Assume: |
'polygon' is one of the following strings discribing the type of curve
to be considered:
|
|||||||||||||||
| Result: | The return value of the procedure is a list of integer vectors. The ith integer vector records to which vertices the ith vertex of the polytope in question is connected. |
14) extremeRaysDB(int,string,string)
| Short: | The procedure returns the extreme rays of the n-th cone in the Gröbner fan of the discriminant respectively the secondary fan. | |||||||||||||||
| Usage: | extremeRaysDB(n,polygon,type); | |||||||||||||||
| Assume: |
'n' is an integer.
'polygon' is one of the following strings discribing the type of curve to be considered:
|
|||||||||||||||
| Result: | The return value of the procedure is an integer matrix whose rows are the extreme rays of the nth cone in the fan in question. |
15) triangulationsDB(int,string)
| Short: | The procedure returns a triangulation of the polygon described by the input. | |||||||||
| Usage: | triangulationsDB(n,polygon,type); | |||||||||
| Assume: |
'n' is an integer.
'polygon' is one of the following strings discribing the type of curve to be considered:
|
|||||||||
| Result: | The return value of the procedure is a list of integer vectors representing the nth triangulation of the polygon in question. |
16) edgeInequalitiesDB(string)
| Short: | The procedure returns the inequalities needed to see a certain edge in a marked subdivision of the polygon in question. | |||||||||
| Usage: | edgeInequalitiesDB(polygon); | |||||||||
| Assume: |
'polygon' is one of the following strings discribing the type of curve
to be considered:
|
|||||||||
| Result: | The return value of the procedure is a list whose first entry is a list of integer matrices such that the ith matrix describes the inequalities which are necessary for the presence of the ith inner edge in a subdivision of the polygon. The second entry is a list where the ith entry consists of the two end points of the ith inner edge. |
17) interiorInequalitiesDB(string)
| Short: | The procedure returns the inequalities needed to see the inner lattice point in a marked subdivision of the polygon in question. | |||||||||
| Usage: | interiorInequalitiesDB(polygon); | |||||||||
| Assume: |
'polygon' is one of the following strings discribing the type of curve
to be considered:
|
|||||||||
| Result: | The return value of the procedure is an integer matrix whose rows represent the inequalities describing the cone such that the interior point of the polygon is a vertex of some polygon in the corresponding marked subdivision of the polygon. |