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If you use the library in your scientific work and want to cite it, please do so as follows:
Anders Nedergaard Jensen, Hannah Markwig, Thomas Markwig: tropical.lib. A SINGULAR 3.0 library for computations in tropical geometry, 2007
In order to be able to use all procedures of this library you should download as well the library
Note, however, that only very few features from that library are used which do NOT invole computations by polymake or topcom. Moreover, if you want to use the procedure tropicalLifting the program gfan must be installed on your system, preferably in version 0.3.0 or higher.
On this web page you will find some information on procedures in the library tropical.lib.
Main procedures in tropical.lib
| Procedures concerned with tropical lifting | ||
|---|---|---|
| • | tropicalLifting | computes a point in the tropical variety of ideal over intvec |
| • | displayTropicalLifting | displays the output of tropicalLifting |
| Procedures concerned with drawing tropical curves | ||
| • | drawTropicalCurve | produces a post script image of the tropical plane curve |
| • | drawNewtonSubdivision | produces a post script image of the Newton subdivision a tropical plane curve |
| • | tropicalCurve | computes a tropical plane curve and its Newton subdivision |
| Procedures conderned with j-invariants | ||
| • | tropicalJInvariant | computes the tropical j-invariant of the tropical curve poly |
| • | weierstrassForm | computes the Weierstrass form of an elliptic curve |
| • | jInvariant | computes the j-invariant of an elliptic curve |
| General procedures | ||
| • | tropicalise | computes the tropicalisation of the polynomial |
| • | tropicaliseSet | computes the tropicalisation of all polynomials in input |
| • | tInitialForm | computes the t-initialform of a polynomial |
| • | tInitialIdeal | computes the t-initial ideal of an ideal |
| • | initialForm | computes the initialform of a polynomial |
| • | initialIdeal | computes the initial ideal of an ideal |
| Procedures concerned with the latex conversion | ||
| • | texNumber | outputs the texcommand for the leading coefficient of poly |
| • | texPolynomial | outputs the texcommand for the polynomial poly |
| • | texMatrix | outputs the texcommand for the matrix |
| • | texDrawBasic | embeds the output of texdrawtropical in a texdraw environment |
| • | texDrawTropical | computes the texdraw commands for a tropical curve |
| • | texDrawNewtonSubdivision | computes the texdraw commands for a Newton subdivision |
| • | texDrawTriangulation | computes the texdraw commands for a triangulation |
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. Do this in the directory where your copy of the library tropical.lib resides. 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 library type
LIB "tropical.lib";
at the prompt. Note, every command in Singular
ends with a semicolon, and note that if you started
SINGULAR
in a directory different from
the one where the library resides, then you have to add the path to the library name.
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| • |
Basically all procedures in this library need polynomials as input. In order to have these at hand
it is necessary do define the polynomial ring in which they life. This will in general either be
Q[t,x1,...,xn], defined by the command (if n=5)
ring r=0,(t,x(1..5)),dp;
or Q(t)[x1,...,xn], defined by the command (if n=5)
ring r=(0,t),(x(1..5)),dp;
If you need a finite field of characteristic p instead, then replace 0 by the prime p.
You can of course also choose other variable names than x(i) -- e.g. x,y,z.
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| • |
You now can define a polynomial respectively an ideal by the commands
poly f=3*x(1)*x(2)^3-t*x(5);
respectively
ideal i=f,x(2)-t*f,x(3)^4-t^3;
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| • | Every object in SINGULAR has its own type and this type has to be specified when defining the object. Besides the types ring, poly, and ideal there are three more types which are needed for this library, namely list defining a list of object of possibly different types, int defining an integer, and intvec defining a vector of integers. | ||||||||||||||
| • |
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| • |
You can invoke the
SINGULAR
help on the library by the command
help tropical.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 tropicalLifting, type
help tropicalLifting;
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 tropicalLifting. 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 ideal), just type
help ideal;
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.
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Details on some procedures
We now want to describe some of the procedures from tropical.lib in more detail.
1) tropicalLifting
| Warning: | The procedure tropicalLifting requires the computer algebra system SINGULAR to be installed on your computer (which is rather obvious), but it also requires the program gfan to be present. Moreover, if your version of gfan is less than 0.3.0, then the procedure MUST be called with the option gfanOld - see below for details on options. If you have SINGULAR version 3-0-2 or less it might be advisable to get a new version of the libraries primdec.lib and absfact.lib from the SINGULAR home page, since some errors have been repaired recently - if you have problems with this please send an email to keilen@math.uni-tuebingen.de. | Short: | The procedure tropicalLifting can be used to lift a point in a tropical variety of an ideal to a point in the algebraic variety of the ideal up to a given order. | ||||||||||||||||||||||||||||||||||
| Long: | Given polynomials f1,...,fk in the polynomial ring Q[t,x1,...,xn] we consider the ideal J generated by the polynomials in C{{t}}[x1,...,xn], the polynomial ring over the field of Puiseux series over the complex numbers. The tropical variety Trop(J) of J is defined to be the negative of the valuation of the points in the the algebraic variety V(J). Since the valuation of a point in C{{t}} is a rational number Trop(J) is a subset of Qn. Given a point w in Trop(J) the procedure computes a point p in V(J) such that val(p)=-w. Of course, since the components of p are Puiseux series, we can only compute them up to a given order. | ||||||||||||||||||||||||||||||||||||
| Usage: |
The procedure needs as input an ideal J, an integer vector w representing the point
in the tropical variety of J, and an integer ord specifying up how many terms of the point
p in V(J) with val(p)=-w should be computed.
Since the point in the tropical variety that we want to lift will in general have rational components, we have to explain how to interprete it as an integer vector. Bring the components to a common denominator and insert this common denominator as first component. E.g. if you want to lift the point (1/2,-1/3) then your input w will be w=6,3,-2. The procedure will produce an output of type list which should be stored for later use and which should best be displayed using the procedure displayTropicalLifting. To find a suitable w in the tropical variety, use gfan. |
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| Example: |
Let us give an example showing how this could be done
in practise.
In this example, the lifting is not a series but it is finite and exact, as the substitution result shows, and no fractional exponents are needed:
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| Result: |
The result of the procedure should be the truncation, say p, of a point in the Puiseux series field.
This, however, is a type which is not available in Singular. Moreover, even though the
coefficients of our input were rational numbers the coefficients of p will lie in a field
extension of Q.
This first of all forces us to create a new ring where the base field is a field extension of Q and which will, basically, contain p. The procedure produces a list (which in the above example was saved as L) and where the first entry is of type ring. You can access this ring (if you have to) by the following sequence of commands: def LIFTRing=LIST[1]; setring LIFTRing; LIFT;
This has created a new ring with name LIFTRing, and in this ring lives the point p with name LIFT.
It is of type ideal and the ith entry encodes the ith component of p.
Since the components of p are Puieseux series which involve
fractional exponents and possibly negative exponents (if w had positive entries).
Both is not available in
SINGULAR
. We get around this problem by the second and third
entry of the list L, which can be viewed by the command:
L[2];L[3];
L[2] is an integer, say N, and the variable t in LIFT should be replaced by t^(1/N);
moreover, if the k+1st entry of L[3] is non-zero, then the kth component of LIFT has to
be multiplied t^(-l[3][k]/l[3][1]) AFTER substituting t by t^1/N. Doing this we finally get
p -- and this should convince you that it is preferable to use displayTropicalLifting
for displaying the result, since it does the transformations for you.
Finally, there is a fourth entry L[4] in the list, which contains a list of strings as entry. The kth string displays the result of substituting p into the kth polynomial in the ideal J. This should give you a feeling if the procedure worked all right. |
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| Options: |
You can call the procedure with a number of options which you can simply add as further arguments
when calling the procedure. The options are as follows:
list L=tropicalLifting(i,w,3,"noAbs","isInTrop");
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| Remark: |
If you want to use other base fields than the rational numbers, like algebraic or transcendental
extensions of Q, you have to simulate these by treating the parameters as variables with weight zero
and possibly adding the relations of the parameters to the ideal J.
You can even use base fields of positive characteristic, like Z/pZ. However, since the Puiseux series field over the algebraic closure of Z/pZ is no longer algebraically closed, there is no guarantee that the algorithm will work - but if you are not too unlucky or if p is sufficiently large, it should work. This might speed up the computations considerable, since many standard basis computations are necessary, and the coefficient growth is a vital problem. However, if you work with positive characteristic then you MUST use the option "noAbs" since the absolute primary decomposition in SINGULAR works only in characteristic zero. |
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| Reference: | The theory and the algorithm are described in the paper: Anders Nedergaard Jensen, Hannah Markwig, Thomas Markwig: An Algorithm for Lifting Points in a Tropical Variety. Preprint (2007). |
2) displayTropicalLifting
| Aim: | The procedure displayTropicalLifting can be used to display the result of tropicalLifting. It takes as input a list which MUST be the output of tropicalLifting. | ||||||||||||||||||||||||
| Result: | None, the procedure does only display something but it has no return value. | ||||||||||||||||||||||||
| Example: |
Let us give an example showing how this could be done
in practise.
In this example, the lifting is not a series but it is finite and exact, as the substitution result shows, and no fractional exponents are needed:
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Option: | The procedure can be called with the additional option "subst", in which case the result of the resubstitution of the lifting into the ideal is shown. See e.g. the example in tropicalLifting. |
3) drawTropicalCurve
| Aim: | The procedure drawTropicalCurve produces a postscript file which displays the tropical f curve defined by a polynomial f in K{{t}}[x,y], i.e. the image of V(f) under the valuation map, and the corresponding Newton subdivision. | ||||||||||||||||
| Note: |
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| Result: | None, the procedure only produces latex and ps files, but it has no return value. | ||||||||||||||||
| Example: |
Let us give an example showing how this could be done
in practise.
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