with plots with LinearAlgebra with Groebner Basis FGLM HilbertDimension HilbertPolynomial HilbertSeries Homogenize InitialForm

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(3) (1) (2) O O (1.1) O O O O with plots : with LinearAlgebra : with Groebner ; Basis, FGLM, HilbertDimension, HilbertPolynomial, HilbertSeries, Homogenize, InitialForm, InterReduce, IsProper, IsZeroDimensional, LeadingCoefficient, LeadingMonomial, LeadingTerm, MatrixOrder, MaximalIndependentSet, MonomialOrder, MultiplicationMatrix, MultivariateCyclicVector, NormalForm, NormalSet, RationalUnivariateRepresentation, Reduce, RememberBasis, SPolynomial, Solve, SuggestVariableOrder, TestOrder, ToricIdealBasis, TrailingTerm, UnivariatePolynomial, Walk, WeightedDegree We consider the following rational map from = to = 2 . F d Vector 6$t 1C t3 , 6$t2 1C t3 ; F := 6 t 1C t3 6 t2 1C t3 The image of this rational map is a parametrized curve known as the Descarte's folium. By a very simple computation show that this curve is included in the curve of implicit equation x3C y3K 6$x$y = 0. 0 Draw the curve x3C y3K 6$x$y using the function implicit plot with the option grid = 100, 100 . The objectives of the two first part of this practical session is to rediscover this result in two different ways if you know the implicit equation in advance.

intersection point

implicit equation

no parametric

groebner basis

xc yc

weierstrass equation

containing only

y2k x3c

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Gröbner basis Elliptic curve

Owith plots:with LinearAlgebra:with Groebner; Basis,FGLM,HilbertDimension,HilbertPolynomial,HilbertSeries,Homogenize,InitialForm,(1) InterReduce,IsProper,IsZeroDimensional,LeadingCoefficient,LeadingMonomial, LeadingTerm,MatrixOrder,MaximalIndependentSet,MonomialOrder, MultiplicationMatrix,MultivariateCyclicVector,NormalForm,NormalSet, RationalUnivariateRepresentation,Reduce,RememberBasis,SPolynomial,Solve, SuggestVariableOrder,TestOrder,ToricIdealBasis,TrailingTerm,UnivariatePolynomial, Walk,WeightedDegree 2 We consider the following rational mapfrom=to=. 2 6$t6$t OFdVector, ; 3 3 1Ct1Ct 6t 3 1Ct F:=(2) 2 6t 3 1Ct The image of this rational mapis a parametrized curve known as the Descarte's folium. By a very simple 3 3 comp utationshow that this curve is included in the curve of implicit equatioxnCyK6$x$y= 0. O0(3) 3 3 Draw the curvexCyK6$x$ytiowith the opp lotusing the function imp licitgnrid100 .= 100, The objectives of the two first part of this practical session is to rediscover this result in two different way sif you know the implicit equation in advance. I) Implicit equation via Groebner basis x If we consider Im(F), it is the set of point suchthat there is a value of parametertsuch that y 2 6$t6$t x=andy= . 3 3 1Ct1Ct 3 32 In other words, Im(F) is contained in the projection ofZ1Ct$xK6$t, 1Ct$yK6$t on thex,yAn ideal is just defined by the list of its generators.-p lane. 3 32 OJd 1Ct$xK6$t, 1Ct$yK6$t; 3 32 J:= 1CtxK6t, 1CtyK6t(1.1) Read the documentation of the function Basis of the Groebner library. Choose an lexicographic order withtOxOyute a Groebner basis ofand compJ. ORemark that the Groebner basis contain an equation containing only the variablesxandyand that this equation is the wanted one. This equation is the first one because Map legives Groebner basis sorted by increasingorder. O

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