Functionalities for genus 2 and 3 curves
aa r X i v : . [ m a t h . AG ] F e b FUNCTIONALITIES FOR GENUS AND CURVES
REYNALD LERCIER, JEROEN SIJSLING, AND CHRISTOPHE RITZENTHALER
Abstract.
We gather and illustrate some functions that we wrote in the
Magma computer algebrasystem for curves of genus and . In genus , we furnish functions both for non-hyperelliptic and forhyperelliptic curves. A fair bit of the functionality in the latter case extends to hyperelliptic curves ofarbitrary genus. This is a technical note to illustrate new functions that we implemented to complete those alreadyincluded in
Magma
Hyperelliptic curves
In this section, we describe functionality for arbitrary genus hyperelliptic curves of genus g > thatare given by a smooth hyperelliptic model y = f ( x ) with f ∈ k [ x ] of degree g + 1 or g + 2 . We assumethe characteristic p of the base field k to be different from .1.1. Computation of isomorphisms.
Let C i : y = f i ( x ) be two hyperelliptic curves of genus g overa base field k . Isomorphisms C → C are of the form ( x, y ) (cid:18) ax + bcx + d , ey ( cx + d ) g +2 (cid:19) (1.1)with (cid:2) a bc d (cid:3) ∈ GL ( k ) and e ∈ k ∗ . (Over an algebraically closed field, one can impose e = 1 , butwe do not insist on this.) The set of isomorphisms is a principal homogeneous space over the groupof automorphisms of either of the curves C and C . Determining if C and C are isomorphic, andreturning the set of isomorphisms if they are, boils down to computing the elements of GL ( k ) whoseright action transforms f into a multiple of f .One possibility for determining these isomorphisms is by applying Gröbner bases after a formal co-efficient comparison, see for example [Gö03]. In [LRS12], we gave alternative algorithms that speed upthis computation when p does not divide g + 2 . Our function IsIsomorphicHyperellipticCurves() determines both the full set of isomorphisms C → C over k itself and that over the algebraic clo-sure of k using the option geometric . We refer to the latter as geometric isomorphisms . In this casethe option commonfield can be enabled to return the list of isomorphisms embedded into a commonoverfield. When working over a number field, this can be an expensive operation. Using the option covariant (which is the default) performs the calculation of these isomorphisms using the reductionprocess involving covariants in [LRS12], whereas disabling this option uses the more direct methodfrom loc. cit . Finally, concentrating on the matrix elements only leads to the definition of reduced isomorphisms or automorphisms, for which we have created dedicated functions as well (for instance ReducedAutomorphismsOfHyperellipticCurve ). Date : February 9, 2021.2010
Mathematics Subject Classification.
Key words and phrases. plane quartic curves, hyperelliptic curves, invariants, reconstruction, descent, twists. xample . We determine the isomorphisms C → C for C : y = x − and C : y = x + 1 . Thecode and the resulting output are as follows. > P
AutomorphismGroup-OfHyperellipticCurve returns a permutation group, followed, if the option explicit is enabled, by anisomorphism to the group of automorphisms of the curve over its base ring or its algebraic closure. Asfor the isomorphism functionality, these group calculations are more efficient than the generic
Magma functionality.
Example . We determine automorphism group of the curve C : y = x − over the finite field F . > P
The computation of representatives for all possible twists of a hyperelliptic curve C : y = f ( x ) over a finite field of characteristic different from is implemented using [CN07], in particular to ruleout so-called self-dual curves . It strongly relies on the computation of the geometric automorphism groupof C . This is implemented by the function Twists() . If the option
AutomorphismGroup is set true , italso outputs the group of reduced automorphisms (i.e. the subgroup of PGL (¯ k ) generated by the firstpart of the representation (1.1) of the geometric automorphisms of C ) as an abstract permutation group.This may currently slow down the algorithm when the group is large, and here there remains room forimprovement. Example . We determine the twists of the hyperelliptic curve C : y = x − over the finite field F . > P
Invariants.
Over an algebraically closed field k of characteristic p with p = 2 , the isomorphismclass of a genus g hyperelliptic curve y = f ( x ) with f ∈ k [ x ] corresponds to the orbit of the binary form z g +2 f ( x/z ) under the classical action of GL ( k ) . One can therefore (see [Dol03, §10.2]) characterize theseclasses by the invariant space Proj( R g ( k )) with R g ( k ) = ( ⊕ n ≥ Sym n (Sym g +2 ( k ))) SL ( k ) . Working outgenerators for the algebra R g ( C ) was a popular pastime of nineteenth-century mathematicians. For g = 2 (that is, for sextic binary forms), this determination goes back to [Cle72], whereas for g = 3 (that is, foroctic binary forms) it goes back to [SF79, VG88].When p = 0 , the situation is more involved. In order to obtain a set of generators for R g ( k ) , agood starting point is often to reduce a set of well-normalized generators of R g ( C ) modulo p . (Herewell-normalized means to be primitive Z -integral polynomials in the coefficients of a generic form; sucha normalization is always possible by [Sil92, Lemma.5.8.1].) Unfortunately, there is currently no way toeasily check whether this reduced set of generators will indeed generate R g ( k ) . There are examples, forinstance g = 3 and p = 5 , where this is not the case.For g = 2 , Igusa [Igu60] managed to give a “universal set of invariants”, which works in every charac-teristic, including . This set of invariants { I , I , I , I , I } (with I being superfluous except when thecharacteristic equals ) is integrated in Magma and can be called by the function
IgusaInvariants() .For g = 3 , thanks to the work of [Gey74], one could show in [LR12] that the reduction of Shiodagenerators for R ( C ) are still generators for R ( k ) when the characteristic of k is greater than . For allsmaller characteristics save p = 5 but including , [Bas15] was able to give a set of separating invariants (i.e., invariants that allow one to separate the orbits of the binary form and therefore to characterize theisomorphism classes). We conjecture that they are also generators. Example . The relevant function for invariants of hyperelliptic curves in genus is ShiodaInvariants .It returns a list I of elements of k to be considered as an element in a weighted projective space with There are also other sets of invariants (IgusaClebschInvariants(), ClebschInvariants()) and absolute invariants(
G2Invariants() ), which are used for historical or practical reasons. Said reference should have included a different proof of [Shi67, Lemma 1], as the one in loc. cit. is only valid incharacteristic . A general proof can be found in [Smi95, Prop.5.5.2]. eights which are indicated by the second output of the function. This list depends on the characteristic p (and when p = 2 on the type defined in [NS04] or [Bas15, Appendix]). When p > , it is a list ofgenerators for R ( k ) of weight (2 , , . . . , . For p = 2 , , , it is a list of separating invariants and for p = 5 a minimal list of invariants that generates the largest subring of invariants that we have beenable to determine so far. Note that the function has a flag PrimaryOnly which only outputs a (proven)homogeneous system of parameters in all characteristic different from .Two lists I can be normalized and compared (and therefore the corresponding curves seen to beisomorphic or not) using the flag normalize:=true , which relies on the techniques of [LR12, Sec.1.4].Another flag is IntegralNormalization , which multiplies the Shioda invariants by certain constants sothat the invariants are defined over Z . One can also choose to get only part of the list of invariants byfiltering them using degmin,degmax . > P
Given the values I of a set of generators for the invariants ofa hyperelliptic curve C : y = f ( x ) over a field k , reconstructing from the invariants means being able toproduce, given only I , a model D : y = g ( x ) that is ¯ k -isomorphic to C . One may try to find D over theground field k itself, and if possible with “small coefficients” when for example k = Q .The general philosophy to reconstruct hyperelliptic curves from the knowledge of their invariants isexplained in [Mes91] and worked out there for a generic genus hyperelliptic curve. It starts with threecovariants of order and then uses beautiful formulas due to Clebsch [Cle72, §103] to construct a conicand a plane curve of degree g + 1 whose intersection are the Weierstrass points of C . In order to find ahyperelliptic model over k , one needs to find a k -rational point for the conic. In the sequel, we will callfields for which Magma can do this computable fields . At any rate, it is easier to produce a model overa quadratic extension of k . For g = 2 , the general case over any computable field (also in characteristic ) was implemented in Magma using the work of [CQ05, CNP05]. The corresponding function is called
HyperellipticCurveFromIgusaInvariants() .In [LR12], this functionality was extended to hyperelliptic curves of genus for computable fields ofcharacteristic p > . Now, thanks to Basson’s work [Bas15], we can also reconstruct over any computablefield of characteristic different from .Note that the reconstruction functions also return the geometric automorphism group of the curveas an abstract permutation group. This involves only the invariants of the curve, as known rela-tions between these describe the locus of curves with a given geometric automorphism group. Alsonote that there exist a function HyperellipticPolynomialFromIgusaInvariants() (resp. a function
HyperellipticPolynomialFromShiodaInvariants() ) that can be applied to the invariants of any GIT-stable sextic (resp. octic binary form). Recall that such a form is defined by having no factor of multi-plicity greater or equal to . Example . We reconstruct a curve from invariants over the base field F . > I := [ GF(3)!1, 0, 0, 0, 0, 0, 1, 0, 1, 2 ];> HyperellipticCurveFromShiodaInvariants(I); yperelliptic Curve defined by y^2 = x^8 + 2 over GF(3)Permutation group acting on a set of cardinality 32(1, 19, 3, 17)(2, 20, 4, 18)(5, 23, 7, 21)(6, 24, 8, 22)(9, 27, 11, 25)(10,28, 12, 26)(13, 31, 15, 29)(14, 32, 16, 30)(1, 9)(2, 10)(3, 11)(4, 12)(5, 13)(6, 14)(7, 15)(8, 16)(17, 29)(18, 30)(19,31)(20, 32)(21, 26)(22, 25)(23, 28)(24, 27) Over Q , we optimized the reconstruction to get models with small integer coefficients. This was alreadyimplemented for g = 3 in [KLL +
18, Sec.3.1] using a tricks involving a so-called “variation of conics” whichfastens the search phase for a rational point on the conic involved in the reconstruction. Oddly a similartrick was harder to develop for g = 2 . The algebra R ( C ) is generated by the invariants I , I , I , I which are algebraically independent and with I such that I ∈ Z [ I , I , I , I ] . The latter is useful toclassify sextic forms under the action of SL ( C ) , yet it becomes irrelevant in the context of classifyingcurves of genus . The reason for this is that the corresponding Proj remains identical when consideringonly the sub-algebra of elements of even degree, that is, sub-algebra Z [ I , I , I , I ] . Therefore, whencomputing the invariants with IgusaInvariants() , I is not stored unless the flag extend is set to true .We could not find a system of four covariants of order to perform variation of conic method whichwould not involve I in the reconstruction process. Fortunately, if I is not provided, there exists arelation of degree 30 that gives I as a function of I , I , I and I . And if by misfortune this relationis not a square over Q , we can substitute λ I , λ I , λ I and λ I for I , I , I , . . . I where λ is aconstant chosen such that λ I is now a square. This yields I up to a sign, which suffices for ourpurposes. (Note that the genus two curves y = f ( x ) and y = − f ( x ) are twists and have the same evendegree Igusa invariants and I ( f ) = − I ( − f ) .)At the end of the procedure, the function MinRedBinaryForm is used to get even smaller coefficients.
Example . We reconstruct a curve of genus from its invariants. > P
For a genus curve over a finite field (including characteristic ), thecomputation of twists has been implemented in all cases. This is now extended similarly to genus using a case-by-case study to work out the explicit coboundaries for specific models determined by theinvariants of the initial curve when the automorphism group is large (or in characteristic ) and by thegeneric method of Section 1.2 for small automorphism groups. xample . We determine the twists of the hyperelliptic curve C : y = x + 1 over F . > P
Dixmier–Ohno invariants.
Isomorphisms of plane smooth quartics over an algebraically closedfield k are induced by linear transformations of the ambient projective plane P . Therefore, isomorphismclasses are characterized by the space Proj( R ( k )) where R ( k ) = ( ⊕ n ≥ Sym n (Sym ( k )) SL ( k ) , i.e. thering of invariants of quartic ternary forms under the classical action of SL ( k ) . When k is of characteristic , Dixmier [Dix87] gave a list of invariants which form a homogeneous system of parameters. It wascompleted by [Ohn07], who furnished a list of generators for the algebra R ( C ) . This invariants arepolynomials in the coefficients of a ternary quartic forms with coefficients in Z [1 / . They can be con-sidered as a point in the weighted projective space with weights (3 , , , , , , , , , , , , .Corresponding functionality was implemented in Magma for the first time in [GK06]. The currentfunction is called
DixmierOhnoInvariants() , and differs from the previous implementations up to somenormalization constants. To recover the initial implementation in [GK06], it suffices to set the flag
IntegralNormalization equal to true . Normalized representatives (for which it suffices to test forequality to decide whether the curves involved are geometrically isomorphic) can also be obtained,namely by setting the flag normalize to true . Example . We consider the Klein quartic and one of its non-trivial twists over Q . > P
A list of generators of the invariants of smooth plane quartics in positive characteristics is not known,although it is suspected that the reduction of the Dixmier–Ohno invariants are generators when thecharacteristic is greater than . In [LLGR20] homogeneous systems of parameters are determined in allcharacteristics except , for which there is a conjectural HSOP that involves an invariant of degree .A call to DixmierOhnoInvariants() in general characteristic outputs a minimal set of invariants thatgenerate the largest subring of invariants that we know of so far.Among the Dixmier–Ohno invariants of a form f ( x, y, z ) , the invariant I of degree plays a par-ticular role. It can be shown that I has integral coefficients and that over any field, its zero locus isprecisely the locus of singular plane quartics. The construction in [GK06] computes the resultant of the3 partial derivatives of f after [GKZ94, p.426]. Unfortunately, this method fails in characteristic forintrinsic reasons, and the original implementation also had issues in characteristic . This is why in thesecharacteristics, we now compute the resultant of the partial derivatives of f with respect to two of thevariables x and y , and of the form f itself. This follows an idea and used programs kindly provided to usby Laurent Busé, which are based on the techniques developed in [BJ14, Def.4.6, Prop.4.7] or [Dem12,Prop.11] and [Jou97, Sec.3.11.19.25]. We obtain a parasitical factor which is the discriminant of thebinary form f | z =0 . However, one can get rid of this issue by deforming the form f into f + ε ( x + y ) incharacteristic different from and f + ε ( x + xy + y ) in characteristic . It then suffices to computethe discriminant of the family thus obtained and to take its value for ε = 0 . Example . We compute the discriminant of the Klein Quartic over F . > P
Given the Dixmier–Ohno invariants I of a generic plane smoothquartic C over a computable field k of characteristic , the algorithms developed in [LRS18] allow thereconstruction of a model of this quartic, which is returned over k itself as long as the geometric automor-phism group of C is not of order . The relevant function is PlaneQuarticFromDixmierOhnoInvariants(I) .We remark the following:(i) In the above, “generic” means concretely that the invariant I is different from . If I iszero, other systems of co- or contra-variants may be chosen to perform the reconstruction.These variants have not been implemented, and there are smooth plane quartics, like the Kleinquartic, for which no such system exists. Regardless, for all non-trivial automorphism strataexcept for the cyclic group Z / Z , as well as for ( Z / Z ) in case I = 0 , an ad hoc reconstructionis performed.(ii) If one would know that the Dixmier–Ohno invariants are generators of R ( k ) , then reconstructionfrom invariants is possible, at least when the characteristic p of k is large enough. Currently, itis not clear when this is the case and the best is to try if the algorithm returns a result (which s then correct). For now we remark that the primes p ≤ or p = 79 are problematic for thegeneric stratum, and that primes up to can be problematic for curves with non-trivialautomorphism group.(iii) If the quartic curve has automorphism group of order , the field of moduli is not necessarily afield of definition and the reconstruction may happen over a quadratic extension only. Still, thealgorithms will in practice often find a model over the field of moduli if it exists.(iv) When k = Q , the variation of conics and the algorithms of [Els09] yield a reconstruction ofquartics with small coefficients, as in [KLL + Example . We reconstruct a plane quartic from its invariants. > P
Isomorphisms.
Isomorphisms and automorphisms of plane quartics have been implemented fol-lowing the covariant method due to van Rijnswou [vR01] that is also used in another form in thereconstruction algorithms. Let C and C be two plane quartic curves over a field k . Our algorithm firstchecks for equality of normalized Dixmier–Ohno invariants of C and C , since if this equality does nothold, no isomorphisms can exist.If this condition is satisfied, the algorithms first try to find the actual isomorphisms C → C underthe assumption that I = 0 . In this case, [vR01] shows that the use of a suitable covariant reduces thisquestion to finding transformations between certain binary forms associated to C and C , which leadsus to the same computation of elements in GL ( k ) that was considered in Section 1.1. In non-genericcases, we have used a direct Gröbner basis method due to Michael Stoll (private communication).Once again the algorithms admit both a version over the base field and a geometric version, withthe latter finding the isomorphisms over the algebraic closure of k . Both versions are very efficient overfinite fields, and the version over the base field is also reasonably fast for k = Q . By contrast, findinggeometric isomorphisms between plane quartic curves over the rationals can still take a fair amount oftime. For more general fields, the implementation still takes too long, and our functions therefore restrictconsiderations to the cases where k is either finite or the rational field. Example . We determine the automorphisms of a plane quartic over the rationals. > P
Magma . As in thehyperelliptic case, they are generally much faster.
Example . We compare timings for our automorphism routine and the native one included in
Magma . > P
Using classical reductions to compute the cohomology set H (Gal(¯ k/k ) , Aut( C )) over finitefields (see for instance [MT10]), we give a function Twists() to compute a list of representatives of alltwists of a smooth plane quartic over a finite field. This relies on the prior computation of the geometricautomorphism group of C . Note that it relies on the function Twists(C, H) that takes as its inputany quasi-projective curve C (not necessarily plane or non-singular) and any finite subgroup H of thegeometric automorphism group C , and that computes the corresponding twists as long as the elementsof H acts as linear transformations of the ambient space, which is for example the case when C iscanonically embedded or smooth. Example . We compute the twists of the Klein quartic over F . > P
For the benefit of the motivated reader, this section lists unanswered questions or functions thatremain to be implemented. The number of stars reflects our naive estimation of the difficulty and/orquantity of the work involved. ⋆ Currently, a generic
Magma functions determines the structure of the reduced automorphismgroup from the list of reduced automorphisms. This stands to be improved, using the classi-fication of reduced automorphism groups. The answer to this question is easier if one is onlyinterested in the abstract group structure, and more complicated if one also wishes to determinea map from this abstract group to the list of reduced automorphisms. ⋆ Find the separants for the invariant ring of binary octic forms in characteristic . ⋆⋆ Prove that the separants for the invariant ring of binary octic forms in characteristic and (and ?) are generators. ⋆⋆ Reconstruct genus hyperelliptic curves from a list of invariants (or separants) in characteristic . ⋆⋆ Prove the correctness of the conjectural HSOP in characteristic . ⋆⋆ Prove that the reductions of the Dixmier-Ohno invariants are still generators for the invariantring if the characteristic of the residue field is larger than . ⋆ ⋆ ⋆ Determine generators for the ring of invariants in smaller characteristic. ⋆ ⋆ ⋆
Make the reconstruction process for generic plane quartics work for all characteristics (or atleast for those greater than ). ⋆ ⋆ ⋆⋆ The same question as the previous one, but this time for all quartics.
References [Bas15] R. Basson.
Arithmétique des espaces de modules des courbes hyperelliptiques de genre en caractéristiquepositive . PhD thesis, Université de Rennes 1, Rennes, 2015.[BJ14] L. Busé and J.-P. Jouanolou. On the discriminant scheme of homogeneous polynomials. Math. Comput. Sci. ,8(2):175–234, 2014. Cle72] A. Clebsch.
Theorie der binären algebraischen formen . Verlag von B.G. Teubner, Leipzig, 1872.[CN07] G. Cardona and E. Nart. Zeta function and cryptographic exponent of supersingular curves of genus 2. In
Pairing-based cryptography—Pairing 2007 , volume 4575 of
Lecture Notes in Comput. Sci. , pages 132–151.Springer, Berlin, 2007.[CNP05] G. Cardona, E. Nart, and J. Pujolàs. Curves of genus two over fields of even characteristic.
Math. Zeitschrift ,250:177–201, 2005.[CQ05] G. Cardona and J. Quer. Field of moduli and field of definition for curves of genus . In Computational aspectsof algebraic curves , volume 13 of
Lecture Notes Ser. Comput. , pages 71–83, Hackensack, NJ„ 2005. World Sci.Publ.[Dem12] M. Demazure. Résultant, discriminant.
Enseign. Math. (2) , 58(3-4):333–373, 2012.[Dix87] J. Dixmier. On the projective invariants of quartic plane curves.
Adv. in Math. , 64:279–304, 1987.[Dol03] I. Dolgachev.
Lectures on invariant theory , volume 296 of
London Mathematical Society Lecture Note Series .Cambridge University Press, Cambridge, 2003.[Els09] A.-S. Elsenhans. Good models for cubic surfaces. Preprint at , 2009.[Gey74] W. D. Geyer. Invarianten binärer Formen. In
Classification of algebraic varieties and compact complex mani-folds , pages 36–69. Lecture Notes in Math., Vol. 412. Springer, Berlin, 1974.[GK06] M. Girard and D. R. Kohel. Classification of genus 3 curves in special strata of the moduli space. In Hess, F.(ed.) et al., Algorithmic number theory. 7th international symposium, ANTS-VII, Berlin, Germany, July 23–28,2006. Proceedings. Berlin: Springer. Lecture Notes in Computer Science 4076, 346-360 (2006)., 2006.[GKZ94] I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky.
Discriminants, resultants, and multidimensional deter-minants . Mathematics: Theory & Applications. Birkhäuser Boston Inc., Boston, MA, 1994.[Gö03] N. Göb. Computing the automorphism groups of hyperelliptic function fields, 2003.[Igu60] J.-I. Igusa. Arithmetic variety of moduli for genus two.
Ann. Math , 72:612–649, 1960.[Jou97] J. P. Jouanolou. Formes d’inertie et résultant: un formulaire.
Adv. Math. , 126(2):119–250, 1997.[KLL +
18] P. Kılıçer, H. Labrande, R. Lercier, C. Ritzenthaler, J. Sijsling, and M. Streng. Plane quartics over Q withcomplex multiplication. Acta Arith. , 185(2):127–156, 2018.[LLGR20] R. Lercier, Q. Liu, E. L. García, and C. Ritzenthaler. Reduction type of smooth quartics, 2020.[LR12] R. Lercier and C. Ritzenthaler. Hyperelliptic curves and their invariants: geometric, arithmetic and algorithmicaspects.
J. Algebra , 372:595–636, 2012.[LRS12] R. Lercier, C. Ritzenthaler, and J. Sijsling. Fast computation of isomorphisms of hyperelliptic curves and explicitdescent. In E. W. Howe and K. S. Kedlaya, editors,
Proceedings of the Tenth Algorithmic Number TheorySymposium , pages 463–486. Mathematical Sciences Publishers, 2012.[LRS18] R. Lercier, C. Ritzenthaler, and J. Sijsling. Reconstructing plane quartics from their invariants.
Discrete &Computational Geometry , pages 1–41, 2018.[LRS20a] R. Lercier, C. Ritzenthaler, and J. Sijsling. hyperelliptic , a
Magma repository for reconstruction and isomor-phisms of hyperelliptic curves. https://github.com/JRSijsling/hyperelliptic , 2020.[LRS20b] R. Lercier, C. Ritzenthaler, and J. Sijsling. quartic , a Magma package for calculating with smooth plane quarticcurves. https://github.com/JRSijsling/quartic , 2020.[Mes91] J.-F. Mestre. Construction de courbes de genre à partir de leurs modules. In Effective methods in algebraicgeometry , volume 94 of
Prog. Math. , pages 313–334, Boston, 1991. Birkäuser.[MT10] S. Meagher and J. Top. Twists of genus three curves over finite fields.
Finite Fields Appl. , 16(5):347–368, 2010.[NS04] E. Nart and D. Sadornil. Hyperelliptic curves of genus three over finite fields of characteristic two.
Finite Fieldsand Their Applications , 10:198–220, 2004.[Ohn07] T. Ohno. The graded ring of invariants of ternary quartics I, 2007. Unpublished.[SF79] J. J. Sylvester and F. Franklin. Tables of the Generating Functions and Groundforms for the Binary Quanticsof the First Ten Orders.
Amer. J. Math. , 2(3):223–251, 1879.[Shi67] T. Shioda. On the graded ring of invariants of binary octavics.
American J. of Math. , 89(4):1022–1046, 1967.[Sil92] J. H. Silverman.
The arithmetic of elliptic curves , volume 106 of
Graduate Texts in Mathematics . Springer-Verlag, New York, 1992. Corrected reprint of the 1986 original.[Smi95] L. Smith.
Polynomial invariants of finite groups , volume 6 of
Research Notes in Mathematics . A K Peters, Ltd.,Wellesley, MA, 1995.[VG88] F. Von Gall. Das vollständige Formensystem der binären Form 7ter Ordnung.
Math. Ann. , 31:318–336, 1888. vR01] S. M. van Rijnswou. Testing the equivalence of planar curves . PhD thesis, Technische Universiteit Eindhoven,Eindhoven, 2001.
Reynald Lercier, DGA & Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France.
Email address : [email protected] Jeroen Sijsling, Institut für Algebra und Zahlentheorie, Universität Ulm, Helmholtzstrasse 18 D-89081Ulm, Germany
Email address : [email protected] Christophe Ritzenthaler, Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France.
Email address : [email protected]@univ-rennes1.fr