Changeset a212fe in git


Ignore:
Timestamp:
Oct 5, 2010, 1:46:34 PM (14 years ago)
Author:
Frank Seelisch <seelisch@…>
Branches:
(u'fieker-DuVal', '117eb8c30fc9e991c4decca4832b1d19036c4c65')(u'spielwiese', '8a337797cc4177aa8747d661d5c4214ea2095dac')
Children:
a1f90e40f6eb4bdcfaf9ef62b43cfc9c02f4f093
Parents:
354d57c31490c4e447bb298f2e25eaaa537479eb
Message:
changes by Wolfram

git-svn-id: file:///usr/local/Singular/svn/trunk@13396 2c84dea3-7e68-4137-9b89-c4e89433aadc
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  • Singular/LIB/paraplanecurves.lib

    r354d57 ra212fe  
    1212OVERVIEW:
    1313
    14 A library for computing rational parametrizations of rational plane curves
    15 defined over Q. @*
    16 Suppose the curve is C = {f(x,y,z)=0} where f is homogeneous of degree
    17 n and absolutely irreducible. (The conditions required will be checked
    18 automatically.) @*
     14Suppose C = {f(x,y,z)=0} is a rational plane curve, where f is homogeneous
     15of degree n with coefficients in Q and absolutely irreducible (these
     16conditions are checked automatically.) @*
    1917After a first step, realized by a projective automorphism in the procedure
    2018adjointIdeal, C satisfies: @*
    2119- C does not have singularities at infinity z=0. @*
    2220- C does not contain the point (0:1:0) (that is, the dehomogenization of f
    23   with respect to the third variable z must be monic as a polynomial in the
    24   second variable y). @*
     21  with respect to z is monic as a polynomial in y). @*
    2522Considering C in the chart z<>0, the algorithm regards x as transcendental
    2623and y as algebraic and computes an integral basis in C(x)[y] of the integral
    2724closure of C[x] in C(x,y) using the  normalization algorithm from
    28 @ref normal.lib: see @ref integralbasis.lib. In a future edition of the library,
    29 also van Hoeij's algorithm for computing the integral basis will be
    30 available. @*
     25@ref{normal.lib}: see @ref{integralbasis.lib}. In a future edition of the
     26library, also van Hoeij's algorithm for computing the integral basis will
     27be available. @*
    3128From the integral basis, the adjoint ideal is obtained by linear algebra.
    3229Alternatively, the algorithm starts with a local analysis of the singular
     
    3431does not correspond to ordinary multiple points or cusps, the integral
    3532basis algorithm is applied separately. The ordinary multiple points and
    36 cusps, in turn, are adressed by a straightforward direct algorithm. The
     33cusps, in turn, are addressed by a straightforward direct algorithm. The
    3734adjoint ideal is obtained by intersecting all ideals obtained locally.
    3835The local variant of the algorithm is used by default. @*
     
    4946REFERENCES:
    5047
     48Janko Boehm: Parametrisierung rationaler Kurven, Diploma Thesis,
     49http://www.math.uni-sb.de/ag/schreyer/jb/diplom%20janko%20boehm.pdf
     50
    5151Theo de Jong: An algorithm for computing the integral closure,
    5252Journal of Symbolic Computation 26 (3) (1998), p. 273-277
     
    5454Gert-Martin Greuel, Santiago Laplagne, Frank Seelisch: Normalization of Rings,
    5555Journal of Symbolic Computation 9 (2010), p. 887-901
    56 
    57 Janko Boehm: Parametrisierung rationaler Kurven, Diploma Thesis,
    58 http://www.math.uni-sb.de/ag/schreyer/jb/diplom%20janko%20boehm.pdf
    5956
    6057Mark van Hoeij: An Algorithm for Computing an Integral Basis in an Algebraic
     
    110107NOTE:   The procedure might fail or give a wrong output if phi does
    111108        not define a birational map.
    112 RETURN: ring, the coordinate ring of P, with an ideal
    113         named J and an ideal named psi.
    114 @format The ideal J defines the image of phi.@*
     109RETURN: ring, the coordinate ring of P, with an ideal named J and an ideal
     110        named psi.@*
     111        The ideal J defines the image of phi.@*
    115112        The ideal psi gives the inverse of phi.@*
    116113        Note that the entries of psi should be considered as representatives
    117114        of classes in the quotient ring R/J.@*
    118 @end format
    119115THEORY: We compute the ideal I(G) in R**S of the graph G of phi.@*
    120116        The ideal J is given by the intersection of I(G) with S.@*
     
    232228///////////////////////////////////////////////////////////////////////////////
    233229proc paraPlaneCurve(poly f, list #)
    234 "USAGE:  paraPlaneCurve(f [,c]); f poly , c optional integer@*
     230"USAGE:  paraPlaneCurve(f [, c]); f poly , c optional integer@*
    235231         optional integer c can be: @*
    236232         1: compute integral basis via normalization. @*
     
    242238         The polynomial f must be homogeneous and absolutely irreducible. @*
    243239         The curve C = {f = 0} must be rational, i.e., have geometric genus 0
    244          (see @ref genus). @*
     240         (see @ref{genus}). @*
     241         These conditions will be checked automatically.
    245242RETURN:  ring with an ideal PARA.
    246243THEORY:  After a first step, realized by a projective automorphism in the
     
    248245- C does not have singularities at infinity z=0. @*
    249246- C does not contain the point (0:1:0) (that is, the dehomogenization of f
    250   with respect to the third variable must be monic as a polynomial in the
    251   second variable). @*
     247  with respect to z is monic as a polynomial in y). @*
    252248Considering C in the chart z<>0, the algorithm regards x as transcendental
    253249and y as algebraic and computes an integral basis in C(x)[y] of the integral
    254 closure of C[x] in C(x,y) using the normalization algorithm from normal.lib
    255 (see integralbasis.lib). In a future edition of the library, also van Hoeij's
     250closure of C[x] in C(x,y) using the normalization algorithm from @ref{normal.lib}:
     251see @ref{integralbasis.lib}. In a future edition of the library, also van Hoeij's
    256252algorithm for computing the integral basis will be available. @*
    257253From the integral basis, the adjoint ideal is obtained by linear algebra.
     
    260256does not correspond to ordinary multiple points or cusps, the integral
    261257basis algorithm is applied separately. The ordinary multiple points and
    262 cusps, in turn, are adressed by a straightforward direct algorithm. The
     258cusps, in turn, are addressed by a straightforward direct algorithm. The
    263259adjoint ideal is obtained by intersecting all ideals obtained locally.
    264260The local variant of the algorithm is used by default. @*
     
    434430///////////////////////////////////////////////////////////////////////////////
    435431proc adjointIdeal(poly f, list #)
    436 "USAGE:  adjointIdeal(f,  [,choices]); f polynomial in three variables, choices
     432"USAGE:  adjointIdeal(f [, choices]); f polynomial in three variables, choices
    437433         optional list consisting of one integer or of one string or of one
    438434         integer followed by one string. @*
    439          optional integer can be: @*
     435         Optional integer can be: @*
    440436         1: compute integral basis via normalization. @*
    441437         2: make local analysis of singularities first and apply normalization
    442438            separately. @*
    443439         The default is 2. @*
    444          optional string may contain substrings: @*
     440         Optional string may contain substrings: @*
    445441         - rattestyes -> causes error message if curve is not rational. @*
    446442         - firstchecksdone -> prevents that check of assumptions will be done
     
    454450transcendental and y as algebraic and computes an integral basis in C(x)[y] of
    455451the integral closure of C[x] in C(x,y) using the normalization algorithm
    456 from normal.lib (see integralbasis.lib). In a future edition of the library,
     452from @ref{normal.lib}: see @ref{integralbasis.lib}. In a future edition of the library,
    457453also van Hoeij's algorithm for computing the integral basis will be available.@*
    458454From the integral basis, the adjoint ideal is obtained by linear algebra.
     
    461457does not correspond to ordinary multiple points or cusps, the integral
    462458basis algorithm is applied separately. The ordinary multiple points and
    463 cusps, in turn, are adressed by a straightforward direct algorithm. The
     459cusps, in turn, are addressed by a straightforward direct algorithm. The
    464460adjoint ideal is obtained by intersecting all ideals obtained locally.
    465461The local variant of the algorithm is used by default. @*
     
    836832         irreducible.
    837833         The plane curve C defined by f is rational.
    838          The ideal I is the adjoint ideal of C.
     834         The ideal AI is the adjoint ideal of C.
    839835RETURN:  ring with an ideal RNC.
    840836EXAMPLE: example mapToRatNormCurve; shows an example
     
    900896proc rncAntiCanonicalMap(ideal I)
    901897"USAGE:  rncAntiCanonicalMap(I); I ideal
    902 ASSUME:  I is a homogeneous ideal in the basering
     898ASSUME:  I is a homogeneous ideal in the basering 
    903899         defining a rational normal curve C in PP^n.
    904900NOTE:   The procedure will fail or give a wrong output if I is not the
     
    944940proc rncItProjOdd(ideal I)
    945941"USAGE:  rncItProjOdd(I); I ideal
    946 ASSUME:  I is a homogeneous ideal in the basering
    947          with n+1 variables defining a rational normal curve C in PP^n with n
    948          odd.
     942ASSUME:  I is a homogeneous ideal in the basering with n+1 variables
     943         defining a rational normal curve C in PP^n with n odd.
    949944NOTE:    The procedure will fail or give a wrong output if I is not the
    950945         ideal of a rational normal curve. It will test whether n is odd.
     
    952947         Note that the entries of PHI should be considered as
    953948         representatives of elements in R/I, where R is the basering.
    954 THEORY:  We iterate the procedure @ref rncAntiCanonicalMap to obtain PHI.
     949THEORY:  We iterate the procedure @ref{rncAntiCanonicalMap} to obtain PHI.
    955950KEYWORDS: rational normal curve, projection.
    956951SEE ALSO: rncItProjEven.
     
    10591054proc rncItProjEven(ideal I)
    10601055"USAGE:  rncItProjEven(I); I ideal
    1061 ASSUME:  I is a homogeneous ideal in the basering
    1062          with n+1 variables defining a rational normal curve C in PP^n with n
    1063          even.
     1056ASSUME:  I is a homogeneous ideal in the basering with n+1 variables
     1057         defining a rational normal curve C in PP^n with n even.
    10641058NOTE:    The procedure will fail or give a wrong output if I is not the
    10651059         ideal of a rational normal curve. It will test whether n is odd.
    1066 RETURN:  ring with an ideal CONIC defining a conic C2 in PP^2. In addition,
    1067          an ideal PHI in the basering defining an isomorphic projection
    1068          of C to C2 will be exported.@*
     1060RETURN:  ring with an ideal CONIC defining a conic C2 in PP^2.@*
     1061         In addition, an ideal PHI in the basering defining an isomorphic
     1062         projection of C to C2 will be exported.@*
    10691063         Note that the entries of PHI should be considered as
    10701064         representatives of elements in R/I, where R is the basering.
    1071 THEORY:  We iterate the procedure @ref rncAntiCanonicalMap to obtain PHI.
     1065THEORY:  We iterate the procedure @ref{rncAntiCanonicalMap} to obtain PHI.
    10721066KEYWORDS: rational normal curve, projection.
    10731067SEE ALSO: rncItProjOdd.
     
    16681662         The polynomial q must be homogeneous of degree 2 and absolutely
    16691663         irreducible. @*
    1670          We have the technical requirement that the basering should neither
    1671          use a variable s nor t.
    16721664NOTE:    The procedure might fail or give a wrong output if the assumptions
    16731665         do not hold.
     
    16771669         parametrization PP^1 --> C2 = {q=0}.
    16781670
    1679 RETURN:  ring with an ideal Ipoint defining a pencil of lines through a point
    1680          on the conic C2 = {q=0}. This point has either coefficients in Q or
    1681          in a quadratic extension field of Q.
    1682 
    1683 THEORY:  We compute a point on C2 via @ref rationalPointConic. The pencil of
     1671THEORY:  We compute a point on C2 via @ref{rationalPointConic}. The pencil of
    16841672         lines through this point projects C2 birationally to PP^1. Inverting
    16851673         the projection gives the result.
     
    23372325ASSUME:  assumes that p is an irreducible quadratic polynomial in the first
    23382326         three ring variables;
    2339          Ground field is expected to be Q.
     2327         ground field is expected to be Q.
    23402328RETURN:  The method finds a point on the given conic. There are two
    23412329         possibilities:
     
    25372525///////////////////////////////////////////////////////////////////////////////
    25382526proc testParametrization(poly f, def rTT)
    2539 "USAGE:  testParametrization(f,rTT); f poly, rTT ring
     2527"USAGE:  testParametrization(f, rTT); f poly, rTT ring
    25402528ASSUME:  The assumptions on the basering and the polynomial f are as required
    2541          by @ref paraPlaneCurve. The ring rTT has two variables and contains
     2529         by @ref{paraPlaneCurve}. The ring rTT has two variables and contains
    25422530         an ideal PARA (such as the ring obtained by applying
    2543          @ref paraPlaneCurve to f).
     2531         @ref{paraPlaneCurve} to f).
    25442532RETURN: int which is 1 if PARA defines a parametrization of the curve
    25452533        {f=0} and 0, otherwise.
    2546 THEORY: We compute the image of PARA and compare it with f.
     2534THEORY: We compute the polynomial defining the image of PARA
     2535        and compare it with f.
    25472536KEYWORDS: Parametrization, image.
    25482537EXAMPLE: example testParametrization; shows an example
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