Changeset 8aa664 in git


Ignore:
Timestamp:
Aug 7, 2008, 5:52:58 PM (16 years ago)
Author:
Thomas Markwig <keilen@…>
Branches:
(u'fieker-DuVal', '117eb8c30fc9e991c4decca4832b1d19036c4c65')(u'spielwiese', 'b52fc4b2495505785981d640dcf7eb3e456778ef')
Children:
21ab56d767a9243f4aace178658aacf6c566ce1f
Parents:
b546c03ad846aa2512dc30210191605ce1f3de85
Message:
the help information and the examples have been reformatted


git-svn-id: file:///usr/local/Singular/svn/trunk@10964 2c84dea3-7e68-4137-9b89-c4e89433aadc
File:
1 edited

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  • Singular/LIB/tropical.lib

    rb546c0 r8aa664  
    1 version="$Id: tropical.lib,v 1.4 2008-08-07 15:02:54 keilen Exp $";
     1version="$Id: tropical.lib,v 1.5 2008-08-07 15:52:58 keilen Exp $";
    22category="Tropical Geometry";
    33info="
    4 LIBRARY:         tropical.lib  Interface to gfan
     4LIBRARY:         tropical.lib  Computations in Tropical Geometry
    55
    66AUTHORS:         Anders Jensen Needergard, email: jensen@math.tu-berlin.de
     
    1010WARNING:
    1111     - tropicalLifting will only work under LINUX and if in addition gfan is installed.
    12      - drawTropicalCurve and drawTropicalNewtonSubdivision will only display the tropical
     12@*   - drawTropicalCurve and drawTropicalNewtonSubdivision will only display the tropical
    1313       curve under LINUX and if in addition latex and kghostview are installed.
    14      - In tropicalLifting the basering needs the parameter t as a variable, while otherwise
     14@*   - In tropicalLifting the basering needs the parameter t as a variable, while otherwise
    1515       it needs it as a parameter.
    1616
     
    4242     i.e. we replace C{{t}} by Q(t), or sometimes even by Q[t]. Note, that this in particular
    4343     forbits rational exponents for the t's.
     44
    4445     Moreover, in Singular no negative exponents of monomials are allowed, so that the integer vectors
    4546     vi will have to have non-negative entries. Shifting all exponents by a fixed integer vector does not
     
    5354
    5455     The main tools provided in this library are as follows:
    55      - tropicalLifting    implements the constructive proof of the Theorem of Newton-Kapranov and constructs
     56@*   - tropicalLifting    implements the constructive proof of the Theorem of Newton-Kapranov and constructs
    5657                          a point in the variety over C{{t}} corresponding to a given point in the
    5758                          corresponding tropical variety associated to an ideal I; the generators of
     
    6061                          several field extensions of Q might be necessary thoughout the intermediate
    6162                          computations; the procedures uses the external program gfan
    62      - drawTropicalCurve  visualises a tropical plane curve either given by a polynomial in Q(t)[x,y]
     63@*   - drawTropicalCurve  visualises a tropical plane curve either given by a polynomial in Q(t)[x,y]
    6364                          or by a list of linear polynomials of the form ax+by+c with a,b in Z and c in Q;
    6465                          latex must be installed on your computer
    65      - tropicalJInvariant computes the tropical j-invaiant of a tropical elliptic curve
    66      - jInvariant         computes the j-invariant of an elliptic curve
    67      - weierstrassFom     computes the Weierstrass form of an elliptic curve     
     66@*   - tropicalJInvariant computes the tropical j-invaiant of a tropical elliptic curve
     67@*   - jInvariant         computes the j-invariant of an elliptic curve
     68@*   - weierstrassFom     computes the Weierstrass form of an elliptic curve     
    6869
    6970PROCEDURES CONCERNED WITH TROPICAL LIFTING:
     
    193194           and ord is the order up to which a point in V(i) over Q{{t}} lying over
    194195           (w_1/w_0,...,w_n/w_0) shall be computed; w_0 may NOT be ZERO
    195          - the basering should not have any parameters on its own
    196            and it should have a global monomial ordering --
    197            e.g. ring r=0,(t,x(1..n)),dp;
    198          - the first variable of the basering will be treated as the parameter t
     196@*       - the basering should not have any parameters on its own
     197           and it should have a global monomial ordering, e.g. ring r=0,(t,x(1..n)),dp;
     198@*       - the first variable of the basering will be treated as the parameter t
    199199           in the Puiseux series field !!!!
    200          - the optional parameter opt should be one or more strings among the following:
    201            'isZeroDimensional'  : the dimension i is zero (not to be checked);
    202            'isPrime'            : the ideal is prime over Q(t)[x_1,...,x_n]  (not to be checked);
    203            'isInTrop'           : (w_1/w_0,...,w_n/w_0) is in the tropical variety (not to be checked);
    204            'oldGfan'            : uses gfan version 0.2.1 or less
    205            'findAll'            : find all solutions of a zero-dimensional ideal over (w_1/w_0,...,w_n/w_0)
    206            'noAbs'              : do NOT use absolute primary decomposition
    207            'noResubst'         : avoids computing the resubstitution
     200@*       - the optional parameter opt should be one or more strings among the following:
     201@*         'isZeroDimensional'  : the dimension i is zero (not to be checked);
     202@*         'isPrime'            : the ideal is prime over Q(t)[x_1,...,x_n]  (not to be checked);
     203@*         'isInTrop'           : (w_1/w_0,...,w_n/w_0) is in the tropical variety (not to be checked);
     204@*         'oldGfan'            : uses gfan version 0.2.1 or less
     205@*         'findAll'            : find all solutions of a zero-dimensional ideal over (w_1/w_0,...,w_n/w_0)
     206@*         'noAbs'              : do NOT use absolute primary decomposition
     207@*         'noResubst'         : avoids computing the resubstitution
    208208RETURN:  IF THE OPTION 'findAll' WAS NOT SET THEN:
    209          list, containing one lifting of the given point (w_1/w_0,...,w_n/w_0)
     209@*       list, containing one lifting of the given point (w_1/w_0,...,w_n/w_0)
    210210               in the tropical variety of i to a point in V(i) over Puiseux series field up to
    211211               the first ord terms; more precisely:
    212                IF THE OPTION 'noAbs' WAS NOT SET THEN:
    213                l[1] = ring Q[a]/m[[t]]
    214                l[2] = int
    215                l[3] = intvec
    216                l[4] = list
    217                IF THE OPTION 'noAbs' WAS SET THEN:
    218                l[1] = ring Q[X(1),...,X(k)]/m[[t]]
    219                l[2] = int
    220                l[3] = intvec
    221                l[4] = list
    222                l[5] = string
    223          IF THE OPITON 'findAll' WAS NOT SET THEN:
    224          list, containing ALL liftings of the given point ((w_1/w_0,...,w_n/w_0)
     212@*             IF THE OPTION 'noAbs' WAS NOT SET THEN:
     213@*             l[1] = ring Q[a]/m[[t]]
     214@*             l[2] = int
     215@*             l[3] = intvec
     216@*             l[4] = list
     217@*             IF THE OPTION 'noAbs' WAS SET THEN:
     218@*             l[1] = ring Q[X(1),...,X(k)]/m[[t]]
     219@*             l[2] = int
     220@*             l[3] = intvec
     221@*             l[4] = list
     222@*             l[5] = string
     223@*       IF THE OPITON 'findAll' WAS NOT SET THEN:
     224@*       list, containing ALL liftings of the given point ((w_1/w_0,...,w_n/w_0)
    225225               in the tropical variety of i to a point in V(i) over Puiseux series field up to
    226226               the first ord terms, if the ideal is zero-dimensional over Q{{t}};
    227227               more precisely, each entry of the list is a list l as computed if
    228228               'find_all' was NOT set
    229          WE NOW DESCRIBE THE LIST ENTRIES IF 'findAll' WAS NOT SET:
    230          - the ring l[1] contains an ideal LIFT, which contains
     229@*       WE NOW DESCRIBE THE LIST ENTRIES IF 'findAll' WAS NOT SET:
     230@*       - the ring l[1] contains an ideal LIFT, which contains
    231231           a point in V(i) lying over w up to the first ord terms;
    232          - and if the integer l[2] is N then t has to be replaced by t^1/N in the
     232@*       - and if the integer l[2] is N then t has to be replaced by t^1/N in the
    233233           lift, or alternatively replace t by t^N in the defining ideal
    234          - if the k+1st entry of l[3] is  non-zero, then the kth component of LIFT has to
     234@*       - if the k+1st entry of l[3] is  non-zero, then the kth component of LIFT has to
    235235           be multiplied t^(-l[3][k]/l[3][1]) AFTER substituting t by t^1/N
    236          - unless the option 'noResubst' was set, the kth entry of list l[4]
     236@*       - unless the option 'noResubst' was set, the kth entry of list l[4]
    237237           is a string which represents the kth generator of
    238238           the ideal i where the coordinates have been replaced by the result of the lift;
    239239           the t-order of the kth entry should in principle be larger than the t-degree of LIFT
    240          - if the option 'noAbs' was set, then the string in l[5] defines a maximal ideal in the
     240@*       - if the option 'noAbs' was set, then the string in l[5] defines a maximal ideal in the
    241241           field Q[X(1),...,X(k)], where X(1),...,X(k) are the parameters of the ring in l[1];
    242242           the basefield of the ring in l[1] should be considered modulo this ideal
    243243REMARK:  - it is best to use the procedure displayTropicalLifting to display the result
    244          - the option 'findAll' cannot be used if 'noAbs' is set
    245          - if the parameter 'findAll' is set AND the ideal i is zero-dimensional in Q{{t}}[x_1,...,x_n]
     244@*       - the option 'findAll' cannot be used if 'noAbs' is set
     245@*       - if the parameter 'findAll' is set AND the ideal i is zero-dimensional in Q{{t}}[x_1,...,x_n]
    246246           then ALL points in V(i) lying over w are computed up to order ord; if the ideal
    247247           is not-zero dimenisonal, then only all points in the ideal after cutting down
    248248           to dimension zero will be computed
    249          - the procedure REQUIRES that the program GFAN is installed on your computer;
     249@*       - the procedure REQUIRES that the program GFAN is installed on your computer;
    250250           if you have GFAN version less than 0.3.0 then you MUST use the optional
    251251           parameter 'oldGfan'
    252          - the procedure requires the Singular procedure absPrimdecGTZ to be present in
     252@*       - the procedure requires the Singular procedure absPrimdecGTZ to be present in
    253253           the package primdec.lib, unless the option 'noAbs' is set; but even if absPrimdecGTZ
    254254           is present it might be necessary to set the option 'noAbs' in order to avoid
     
    256256           extension which is computed throughout the recursion might need more only one
    257257           parameter to be described
    258          - since Q is infinite, the procedure finishes with probability one
    259          - you can call the procedure with Z/pZ as base field instead of Q, but there
     258@*       - since Q is infinite, the procedure finishes with probability one
     259@*       - you can call the procedure with Z/pZ as base field instead of Q, but there
    260260           are some problems you should be aware of:
    261            - the Puiseux series field over the algebraic closure of Z/pZ is NOT algebraicall closed,
     261@*         + the Puiseux series field over the algebraic closure of Z/pZ is NOT algebraicall closed,
    262262             and thus there may not exist a point in V(i) over the Puiseux series field
    263263             with the desired valuation; so there is no chance that the procedure produced
    264264             a sensible output - e.g. if i=tx^p-tx-1
    265            - if the dimension of i over Z/pZ(t) is not zero the process of reduction to
     265@*         + if the dimension of i over Z/pZ(t) is not zero the process of reduction to
    266266             zero might not work if the characteristic is small and you are unlucky
    267            - the option 'noAbs' has to be used since absolute primary decomposition in
     267@*         + the option 'noAbs' has to be used since absolute primary decomposition in
    268268             Singular only works in characteristic zero
    269          - the basefield should either be Q or Z/pZ for some prime p; field extensions
     269@*       - the basefield should either be Q or Z/pZ for some prime p; field extensions
    270270           will be computed where necessary; if you need parameters or field extensions
    271271           from the beginning they should rather be simulated as variables possibly adding
     
    717717RETURN:      none
    718718NOTE:        - the procedure displays the output of the procedure tropicalLifting
    719              - if the optional parameter 'subst' is given, then the lifting is
     719@*           - if the optional parameter 'subst' is given, then the lifting is
    720720               substituted into the ideal and the result is displayed
    721721EXAMPLE:     example displayTropicalLifting;   shows an example"
     
    11481148// the coordinates of the first vertex are graph[1][1],graph[1][2];
    11491149   graph[1][1],graph[1][2];
    1150 // the first vertex is connected to the vertices graph[1][3][1,1..ncols(graph[1][3])]
     1150// the first vertex is connected to the vertices
     1151//     graph[1][3][1,1..ncols(graph[1][3])]
    11511152   intmat M=graph[1][3];
    11521153   M[1,1..ncols(graph[1][3])];
    1153 // the weights of the edges to these vertices are graph[1][3][2,1..ncols(graph[1][3])]
     1154// the weights of the edges to these vertices are
     1155//     graph[1][3][2,1..ncols(graph[1][3])]
    11541156   M[2,1..ncols(graph[1][3])];
    11551157// from the first vertex emerge size(graph[1][4]) unbounded edges
    11561158   size(graph[1][4]);
    1157 // the primitive integral direction vector of the first unbounded edge of the first vertex
     1159// the primitive integral direction vector of the first unbounded edge
     1160//     of the first vertex
    11581161   graph[1][4][1][1];
    11591162// the weight of the first unbounded edge of the first vertex
    11601163   graph[1][4][1][2];
    1161 // the monomials in f which are part of the Newton subdivision of the first vertex
     1164// the monomials which are part of the Newton subdivision of the first vertex
    11621165   graph[1][5];
    1163 // connecting the points in graph[size(graph)][1] we get the boundary of the Newton polytope
     1166// connecting the points in graph[size(graph)][1] we get
     1167//     the boundary of the Newton polytope
    11641168   graph[size(graph)][1];
    1165 // an entry in graph[size(graph)][2] is a pair of points in the Newton polytope bounding an inner edge
     1169// an entry in graph[size(graph)][2] is a pair of points
     1170//     in the Newton polytope bounding an inner edge
    11661171   graph[size(graph)][2][1];
    11671172}
     
    11781183               /tmp/tropicalcurveNUMBER.ps, where NUMBER is a random four digit integer;
    11791184               moreover it displays the tropical curve defined by f via kghostview;
    1180              - edges with multiplicity greater than one carry this multiplicity
    1181              - if # is empty, then the tropical curve is computed w.r.t. minimum, if #[1] is the
     1185@*           - edges with multiplicity greater than one carry this multiplicity
     1186@*           - if # is empty, then the tropical curve is computed w.r.t. minimum, if #[1] is the
    11821187               string 'max', then it is computed w.r.t. maximum
    1183              - if the last optional argument is 'onlytexfile' then only the latex file
     1188@*           - if the last optional argument is 'onlytexfile' then only the latex file
    11841189               is produced; this option should be used if kghostview is not installed on
    11851190               your system
    1186              - note that lattice points in the Newton subdivision which are black correspond to markings
     1191@*           - note that lattice points in the Newton subdivision which are black correspond to markings
    11871192               of the marked subdivision, while lattice points in grey are not marked
    11881193EXAMPLE:     example drawTropicalCurve  shows an example"
     
    13341339            where NUMBER is a random four digit integer;
    13351340            moreover it desplays the tropical curve defined by f via kghostview;
    1336             if # is empty, then the tropical curve is computed w.r.t. minimum, if #[1] is the
     1341@*          if # is empty, then the tropical curve is computed w.r.t. minimum, if #[1] is the
    13371342            string 'max', then it is computed w.r.t. maximum
    1338           - note that lattice points in the Newton subdivision which are black correspond to markings
     1343@*        - note that lattice points in the Newton subdivision which are black correspond to markings
    13391344            of the marked subdivision, while lattice points in grey are not marked
    13401345EXAMPLE:     example drawNewtonSubdivision;   shows an example"
     
    14191424             alternatively f can be a polynomial in Q(t)[x,y] defining a tropical plane curve
    14201425             via the valuation map;
    1421              the basering must have a global monomial ordering, two variables and up to one parameter!
     1426@*           the basering must have a global monomial ordering, two variables and up to one parameter!
    14221427RETURN:      number, if the graph underlying the tropical curve has precisely one loop then its weighted
    14231428                     lattice length is returned, otherwise the result will be -1
    14241429NOTE:        - if the tropical curve is elliptic and its embedded graph has precisely one loop,
    14251430               then the weigthed lattice length of the loop is its tropical j-invariant
    1426              - the procedure checks if the embedded graph of the tropical curve has genus one,
     1431@*           - the procedure checks if the embedded graph of the tropical curve has genus one,
    14271432               but it does NOT check if the loop can be resolved, so that the curve is not
    14281433               a proper tropical elliptic curve
    1429              - if the embedded graph of a tropical elliptic curve has more than one loop, then
     1434@*           - if the embedded graph of a tropical elliptic curve has more than one loop, then
    14301435               all but one can be resolved, but this is not observed by this procedure, so it
    14311436               will not compute the j-invariant
    1432              - if # is empty, then the tropical curve is computed w.r.t. minimum, if #[1] is the
     1437@*           - if # is empty, then the tropical curve is computed w.r.t. minimum, if #[1] is the
    14331438               string 'max', then it is computed w.r.t. maximum
    1434              - the tropicalJInvariant of a plane tropical cubic is the 'cycle length' of the
     1439@*           - the tropicalJInvariant of a plane tropical cubic is the 'cycle length' of the
    14351440               cubic as introduced in the paper:
    14361441               Erik Katz, Hannah Markwig, Thomas Markwig: The j-invariant of a cubic tropical plane curve.
     
    15781583   echo=2;
    15791584   ring r=(0,t),(x,y),dp;
    1580 // tropcial_j_invariant computes the tropical j-invariant of the elliptic curve f
     1585// tropcialJInvariant computes the tropical j-invariant of an elliptic curve
    15811586   tropicalJInvariant(t*(x3+y3+1)+1/t*(x2+y2+x+y+x2y+xy2)+1/t2*xy);
    15821587// the Newton polygone need not be the standard simplex
     
    15841589// the curve can have arbitrary degree
    15851590   tropicalJInvariant(t*(x7+y7+1)+1/t*(x4+y4+x2+y2+x3y+xy3)+1/t7*x2y2);
    1586 // the procedure does not realise, if the embedded graph of the tropical curve has
    1587 // a loop that can be resolved
     1591// the procedure does not realise, if the embedded graph of the tropical
     1592//     curve has a loop that can be resolved
    15881593   tropicalJInvariant(1+x+y+xy+tx2y+txy2);
    15891594// but it does realise, if the curve has no loop at all ...
    15901595   tropicalJInvariant(x+y+1);
    1591 // or if the embedded graph has more than one loop - even if only one cannot be resolved
     1596// or if the embedded graph has more than one loop - even if only one
     1597//     cannot be resolved
    15921598   tropicalJInvariant(1+x+y+xy+tx2y+txy2+t3x5+t3y5+tx2y2+t2xy4+t2yx4);
    15931599}
     
    16011607NOTE:        - the algorithm for the coefficients of the Weierstrass form is due to
    16021608               Fernando Rodriguez Villegas, villegas@math.utexas.edu
    1603              - the characteristic of the base field should not be 2 or 3
    1604              - if an additional argument # is given, a simplified Weierstrass form
     1609@*           - the characteristic of the base field should not be 2 or 3
     1610@*           - if an additional argument # is given, a simplified Weierstrass form
    16051611               is computed
    16061612EXAMPLE:     example weierstrassForm;   shows an example"
     
    16471653   poly wg=weierstrassForm(g);
    16481654   wg;
    1649 // ... but it is not yet a simple, since it still has an xy-term, unlike swg
     1655// but it is not yet a simple, since it still has an xy-term, unlike swg
    16501656   poly swg=weierstrassForm(g,1);
    16511657   swg;
    1652 // the j-invariants of all three polynomials coincide ...
     1658// the j-invariants of all three polynomials coincide
    16531659   jInvariant(g);
    16541660   jInvariant(wg);
     
    16651671               point, so that it defines an elliptic curve on the toric surface corresponding
    16661672               to the Newton polygon
    1667              - it the optional argument # is present the base field should be Q(t) and
     1673@*           - it the optional argument # is present the base field should be Q(t) and
    16681674               the optional argument should be one of the following strings:
    1669                'ord'   : then the return value is of type integer, namely the order of the j-invariant
    1670                'split' : then the return value is a list of two polynomials, such that the quotient
     1675@*             'ord'   : then the return value is of type integer, namely the order of the j-invariant
     1676@*             'split' : then the return value is a list of two polynomials, such that the quotient
    16711677                         of these two is the j-invariant
    16721678RETURN:      poly, the j-invariant of the elliptic curve defined by poly
     
    16961702   echo=2;
    16971703   ring r=(0,t),(x,y),dp;
    1698 // jInvariant computes the j-invariant of a cubic independent of the ground field
     1704// jInvariant computes the j-invariant of a cubic
    16991705   jInvariant(x+y+x2y+y3+1/t*xy);
    1700 // if the ground field has one parameter t, then we can instead compute the order of the j-invariant
     1706// if the ground field has one parameter t, then we can instead
     1707//    compute the order of the j-invariant
    17011708   jInvariant(x+y+x2y+y3+1/t*xy,"ord");
    17021709// one can compare the order of the j-invariant to the tropical j-invariant
     
    17181725ASSUME:      points is a list of five points in the plane over K(t)
    17191726RETURN:      list, l[1] = the list points of the five given points
    1720                    l[2] = the conic f passing through the five points
    1721                    l[3] = list of equations of the tangents to f in the given points
    1722                    l[4] = ideal, the tropicalisation of f (i.e. a list of linear forms)
    1723                    l[5] = a list of the tropicalisation of the tangents
    1724                    l[6] = a list containing the vertices of the tropical conic f
    1725                    l[7] = a list containing lists with the vertices of the tangents
    1726                    l[8] = a string which contains the latex-code to draw the tropical
     1727@*                 l[2] = the conic f passing through the five points
     1728@*                 l[3] = list of equations of the tangents to f in the given points
     1729@*                 l[4] = ideal, the tropicalisation of f (i.e. a list of linear forms)
     1730@*                 l[5] = a list of the tropicalisation of the tangents
     1731@*                 l[6] = a list containing the vertices of the tropical conic f
     1732@*                 l[7] = a list containing lists with the vertices of the tangents
     1733@*                 l[8] = a string which contains the latex-code to draw the tropical
    17271734                          conic and its tropicalised tangents
    1728                    l[9] = if # is non-empty, this is the same data for the dual conic
     1735@*                 l[9] = if # is non-empty, this is the same data for the dual conic
    17291736                          and the points dual to the computed tangents
    17301737NOTE:        the points must be generic, i.e. no three on a line
     
    19111918// conic[2] is the equation of the conic f passing through the five points
    19121919   conic[2];
    1913 // conic[3] is a list containing the equations of the tangents through the five points
     1920// conic[3] is a list containing the equations of the tangents
     1921//          through the five points
    19141922   conic[3];
    19151923// conic[4] is an ideal representing the tropicalisation of the conic f
    19161924   conic[4];
    1917 // conic[5] is a list containing the tropicalisation of the five tangents in conic[3]
     1925// conic[5] is a list containing the tropicalisation
     1926//          of the five tangents in conic[3]
    19181927   conic[5];
    1919 // conic[6] is a list containing the vertices of the tropical conic defined by f
     1928// conic[6] is a list containing the vertices of the tropical conic
    19201929   conic[6];
    19211930// conic[7] is a list containing the vertices of the five tangents
    19221931   conic[7];
    1923 // conic[8] contains the latex code to draw the tropical conic and its tropicalised tangents;
    1924 // it can written in a file, processed and displayed via kghostview -- to do so hit the return button
     1932// conic[8] contains the latex code to draw the tropical conic and
     1933//          its tropicalised tangents; it can written in a file, processed and
     1934//          displayed via kghostview
    19251935   write(":w /tmp/conic.tex",conic[8]);   
    1926    system("sh","cd /tmp; latex /tmp/conic.tex; dvips /tmp/conic.dvi -o; kghostview conic.ps &");
    1927 // with an optional argument the same information for the dual conic is computed and saved in conic[9]
     1936   system("sh","cd /tmp; latex /tmp/conic.tex; dvips /tmp/conic.dvi -o;
     1937            kghostview conic.ps &");
     1938// with an optional argument the same information for the dual conic is computed
     1939//         and saved in conic[9]
    19281940   conic=conicWithTangents(points,1);
    19291941   conic[9][2]; // the equation of the dual conic
     
    19391951RETURN:      list, the linear forms of the tropicalisation of f
    19401952NOTE:        if # is empty, then the valuation of t will be 1,
    1941              if # is the string 'max' it will be -1; the latter
    1942             supposes that we consider the maximum of the the computed
     1953@*           if # is the string 'max' it will be -1;
     1954@*           the latter supposes that we consider the maximum of the the computed
    19431955             linear forms, the former that we consider their minimum
    19441956EXAMPLE:     example tropicalise;   shows an example"
     
    23692381           insert further texdraw commands (e.g. to have a lighter image as when called
    23702382           from inside conicWithTangents);
    2371          - the procedure computes a scalefactor for the texdraw command which should
     2383@*       - the procedure computes a scalefactor for the texdraw command which should
    23722384           help to display the curve in the right way; this may, however, be a bad idea
    23732385           if several texDrawTropical outputs are put together to form one image; the
    23742386           scalefactor can be prescribed by a second optional entry of type poly
    2375          - the list # is optional and may as well be empty
     2387@*       - the list # is optional and may as well be empty
    23762388EXAMPLE:     example texDrawTropical;   shows an example"
    23772389{
     
    25342546               insert further texdraw commands (e.g. to have a lighter image as when called
    25352547               from inside conicWithTangents); the list # is optional and may as well be empty
    2536              - note that lattice points in the Newton subdivision which are black correspond to markings
     2548@*           - note that lattice points in the Newton subdivision which are black correspond to markings
    25372549               of the marked subdivision, while lattice points in grey are not marked
    25382550EXAMPLE:     example texDrawNewtonSubdivision;   shows an example"
     
    26662678   poly f=x+y+x2y+xy2+1/t*xy;
    26672679   list graph=tropicalCurve(f);
    2668 // compute the texdraw code of the Newton subdivision of the tropical curve defined by f
     2680// compute the texdraw code of the Newton subdivision of the tropical curve
    26692681   texDrawNewtonSubdivision(graph);
    26702682}
     
    27512763   // the lattice polygon spanned by the points (0,0), (3,0) and (0,3)
    27522764   // with all integer points as markings
    2753    list polygon=intvec(1,1),intvec(3,0),intvec(2,0),intvec(1,0),intvec(0,0),intvec(2,1),intvec(0,1),intvec(1,2),intvec(0,2),intvec(0,3);
    2754    // define a triangulation by connecting the only interior point with the vertices
     2765   list polygon=intvec(1,1),intvec(3,0),intvec(2,0),intvec(1,0),intvec(0,0),
     2766                intvec(2,1),intvec(0,1),intvec(1,2),intvec(0,2),intvec(0,3);
     2767   // define a triangulation by connecting the only interior point
     2768   //        with the vertices
    27552769   list triang=intvec(1,2,5),intvec(1,5,10),intvec(1,2,10);
    27562770   // produce the texdraw output of the triangulation triang
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