Rationality and Parametrizations of Algebraic Curves under Specializations

Rational algebraic curves have been intensively studied in the last decades, both from the theoretical and applied point of view. In applications (e.g. level curves, linear homotopy deformation, geometric constructions in computer aided design, etc.), there often appear unknown parameters. It is possible to adjoin these parameters to the coefficient field as transcendental elements. In some particular cases, however, the curve has a different behavior than in the generic situation treated in this way. In this paper, we show when the singularities and thus the (geometric) genus of the curves might change. More precisely, we give a partition of the affine space, where the parameters take values, so that in each subset of the partition the specialized curve is either reducible or its genus is invariant. In particular, we give a Zariski-closed set in the space of parameter values where the genus of the curve under specialization might decrease or the specialized curve gets reducible. For the genus zero case, and for a given rational parametrization, a better description is possible such that the set of parameters where Hilbert's irreducibility theorem does not hold can be isolated, and such that the specialization of the parametrization parametrizes the specialized curve. We conclude the paper by illustrating these results by some concrete applications.


Introduction
The study and analysis of the behavior of algebraic or algebraic-geometric objects under specializations is of great interest from a theoretical, computational or applied point of view.For instance, some techniques for computing resultants, gcds, or polynomial factorizations, rely on Hensel's lemma or the Chinese remainder theorem (see e.g.[11], [38]).From a more theoretical point of view, also computational, it is important to control, for instance, when a resultant, or more generally a Gröbner basis with parameters, specializes properly (see e.g.[4], [17]).The question whether a given irreducible polynomial over K(a 1 , . . ., a n ) remains irreducible when the parameters are replaced by values in a field K was studied intensively by Hilbert [6] and Serre [30] and is the defining property of "Hilbertian fields".The work of Serre can be seen in a more general context.Another clear example is the resolution of problems in matrix analysis.Besides the basic analysis of systems with parameters, this type of situation appears in the computation of generalized functional inverses (see e.g.[24]).
In relation to the applications derived from the particular results of this paper, the following can be mentioned.Geometric constructions in computer aided design, like offsets, conchoids, cissoids etc., introduce parameters as the distance, the matrix entries of an isometry, or the coordinates of the foci; see e.g.[1], [16], [18], [23].Usually it is interesting to analyze geometric properties of these varieties under the specialization of the parameters; for instance, the genus of offsets is analyzed in [2], and the rationality of the conchoids depending on the foci in [23].Another possible application is the analysis of the rationality of the level curves of surfaces, where one of the variables is taken as a parameter, or the linear homotopy deformation of curves.Curve recognition, via Hough transform (see e.g.[33]), is another field of potential applicability of the results.In this context, a catalog of curves, i.e. a family of curves depending on several parameters, as well as a cloud of points, are given.Then, parameter values providing the best approximation, among the catalog, of the cloud of points, is computed.In this frame one may think on determining rational curve solutions in the catalog.Finally, one may mention the rational solutions of functional algebraic equations, or the solution of algebraic differential equations which coefficients depend on parameters (see [8]).In Section 7, we illustrate some of these applications by means of examples.
In this paper, we study algebraic curves C(F ) given as the zero-set of a polynomial (1.1) F (x, y) = 0 with F ∈ K(a 1 , . . ., a n )[x, y] where K is a computable field of characteristic zero, a 1 , . . ., a n are a set of parameters, and F is irreducible over the algebraic closure of the coefficient field K(a 1 , . . ., a n ).We focus on the problem that for certain values of the parameters a 1 , . . ., a n the algebraic properties of the resulting curve do not coincide with the generic properties of C(F ).More precisely, we define several Zariski-closed sets in the space of parameter values where non-generic behavior may appear.Of particular interest are the singularities, their multiplicities and their character.This leads to a partition of the affine space, where the parameters take values, so that in each subset of the partition the specialized curve is either reducible or its (geometric) genus is invariant.When the generic curve has genus zero, and a rational parametrization, depending on the parameters a 1 , . . ., a n , is given, we provide a new sub-partition where the parametrization specializes properly.In particular, the set of parameters where Hilbert's irreducibility theorem does not hold can be identified.Moreover, the birationality of the specialization of the rational parametrization is guaranteed.To the best of our knowledge, this is the first systematic computational study of the problem.
In [36,10,3,29], and references therein, algebraic curves and their rationality are studied.The problem of finding rational parametrizations of plane curves is a classical problem and has already been studied by Hilbert and Hurwitz [7], and more recently in [13], [20], [26], [27].In addition, for evaluating the parameters, it is important to control field extensions which might be necessary for computing parametrizations.Optimal fields of parametrizations have been studied in [13] and [27].When introducing parameters in the coefficients, new phenomena have to be considered and lead to Tsen's study of finding solutions in a minimal field [5].
The structure of the paper is as follows.In Section 2 we present some preliminaries on algebraic curves and rational parametrizations.In Section 3, we introduce the unspecified parameters and their specialization.The computation of the genus and rational parametrizations is followed to define several computable Zariski-open subsets Ω where the specialized curve behaves, up to irreducibility, as in the generic case.The actual computation of the genus is presented in Section 4. In Theorem 4.2 is shown that the genus of the specialized curve, where the parameters take values in Ω singOrd , is less or equal to the generic genus or the defining polynomial is reducible.A direct corollary of that is that specialized curves of rational curves are also rational or reducible (Corollary 4.3).For values in a smaller set Ω genusOrd , it is shown that the genus of the curve remains exactly the same, again up to irreducibility, see Theorem 4.10.Section 5 is devoted to the case where the generic curve is rational; in this frame the irreducibility can be guaranteed.For some of the parameter values the genus may remain the same but an evaluation of the parametrization is not possible.In Theorem 5.5, however, is presented an open set where the specialization is possible and results in a parametrization of the specialized curve.These open sets can be recursively used for decomposing the whole parameter space as it is explained in Section 6. Applications as described above are presented by using illustrative examples in Section 7.
The decomposition of the parameter space, derived from our analysis, depends on the particular method used to compute the genus as well as the rational parametrization.There exist different methods to deal computationally with the genus: the adjoint curve based method (see e.g.[36], [3] and [29]), the method based on the anticanonical divisor (see [13]) or the method based on Puiseux expansions (see [19]), among others.Similarly for the rational parametrization.In this paper we will follow the adjoint curve based method.For the sake of completeness of the paper, we have included an appendix oriented to give the necessary computational details on how the genus and parametrizations are computed.
This manuscript is a self-contained work on the computation of the genus and rational parametrizations of algebraic curves involving parameters.Results from various mathematical disciplines are combined for this purpose and presented in a coherent way.A rigorous construction of such computable Zariski-open sets were, up to our knowledge, missing in the literature.The theorems mentioned in the previous paragraph are novel and can be directly applied in several interesting problems involving parametric curves.
Notation.Throughout this paper, the following notation will be used.
• We denote by a a tuple of parameters.
• K is a computable field of characteristic zero.We represent by L the field extension L ∶= K(a).In addition, we consider an algebraic element γ over L. Let F be the field F ∶= L(γ).Furthermore, K represents any field extension of K. We denote by K the algebraic closure of K, similarly for any field appearing in the paper.• S is the affine space where a will take values.• For G ∈ K[x, y] ∖ K, we denote by C(G) the plane affine algebraic curve We denote by G h (x, y, z) the homogenization of G, and by G x , G y (similarly for G h x , G h y , G h z ) the partial derivative of G w.r.t.x and y respectively.For a homogeneous polynomial M ∈ K[x, y, z] ∖ {0}, C(M ) denotes the projective plane curve (1.4) • For polynomials f, g in the variable t, and coefficients in an integral domain, we denote by res t (f, g) the resultant of f and g w.r.t.t.
where v is a tuple of variables.We denote by V(f 1 , . . ., f k ) the common zero set, over K, of the polynomials {f 1 , . . ., f k }; similarly for V(I) where where v is a tuple of variables, be a polynomial.We denote by deg(R) the total degree of R w.r.t.v.For a particular variable v i in v, we denote by , where v is a tuple of variables, be a rational function.We denote by num(R) and denom(R) the numerator and denominator of R, respectively, assuming that R is expressed in reduced form; that is, the numerator and denominator are assumed to be coprime.We denote by deg(R) ∶= max{deg(num(R)), deg(denom(R))}, and

Preliminaries on Rational Curves
In this section we recall some notions and results related to rational (plane) curves that we will be important throughout the paper; for further details we refer to [29].In addition, some more technical issues, needed in the development of the paper, are included in the appendix.For this purpose, throughout this section, let G ∈ K[x, y] ∖ K.
2.1.Rational Curves.The zero set C(G) of G, over the algebraic closure K, is called the affine plane curve associated to G (see (1.3)).Taking the homogenization G h of G, we associate to C(G) the projective plane curve C(G h ), see (1.4).Then G (resp.
The degree of C(G) (similarly for C(G h )) is defined as the total degree of the polynomial G and we denote it as deg(C(G)).Moreover, we say that C(G) is irreducible (similarly for and irreducibility of C(G) and C(G h ) are equivalent.In the sequel, we assume that C(G) is irreducible.Let us recall that p ∈ C(G) is a singular point of C(G) of multiplicity r ∈ N, also called r-fold point, if all partial derivatives of G of order at most r − 1 vanish at p and, at least, one partial derivative of order r does not vanish.The tangents to C(G) at an r-fold point p ∈ C(G) are introduced as the lines defined by the linear factors of the sum of all terms in the Taylor expansion of G around p generated by the partial derivatives of G, at p, of order r.If all tangents at p are different, p is called ordinary otherwise is called non-ordinary.We call character of a singularity, the fact of being ordinary or non-ordinary.The notions of singularity, tangent, and character, are similarly introduced for C(G h ).In the appendix we see how to compute and manipulate the singular locus of the curve.
For the purposes of the paper, the notion of rational parametrization is important.A rational (affine) parametrization of C(G) is a pair where p i are co-prime homogeneous forms, of the same degree, over K, not all zero, such that G h (Q) = 0.Not all curves can be rationally parametrized.Curves admitting a rational parametrization are called rational curves.Rational curves can be characterized by means of the genus of the curve.Intuitively speaking the genus measures the difference between the maximum number of singularities (properly counted) the curve, of a fixed degree, may have, and the actual number of singularities it does have.More formally, for an irreducible projective curve C(G h ), the (geometric) genus can be defined in various ways, for instance, via the linear space of divisors; see [29,Definition 3.4], [10,Section 8.6] or [3,Section 8.2] for more details.The genus of C(G) is defined as the genus of C(G h ).Now rational curves are precisely the irreducible curves of genus zero (see e.g.Theorem 4.63 in [29] or [3, page 210]).As commented in the introduction, the genus can be computed in different ways.There exist algorithmic methods to compute the genus of an algebraic curve and to determine, when the genus is zero, a rational parametrization of the curve (see e.g.[13], [20], [26], [27], [29]).Within this paper, we make use of the adjoint curves based method for parametrizing curves which is summarized in Appendix A.

Fields of Parametrization.
In general, if one computes a parametrization P(t) of C(G), the ground field, that is the smallest field where the coefficients of G belong to, has to be extended (see e.g.Sections 4.7.and 4.8. in [29]).A field is called a parametrizing field or field of parametrization of C(G) if there exists a parametrization of C(G) with coefficients in it.In this subsection, we recall some results on fields of parametrization.
Hilbert-Hurwitz Theorem (see [7]) plays a fundamental role in this context.Thus, let us briefly comment it.For this purpose, we will use the notion of adjoint curve (see e.g.Definition 4.55 and 4.56 in [29] or Subsection A.4 in the appendix).Given a projective rational plane curve C(G h ) of degree d, Hilbert-Hurwitz Theorem states that almost all 3tuples of (d − 2)-degree adjoints, to C(G h ), define a rational map from P 2 to P 2 , that maps birationally C(G h ) onto a (d − 2)-degree projective irreducible plane curve, say C(G * ).Now, since the ground field of the original curve and that of the adjoints coincides, say K, (see e.g.Theorem 4.66 in [29]), the ground field of C(G * ) is also K. Furthermore, since birational maps preserve the genus of the curve (see e.g.[3, Theorem 7.2.2 or Prop.8.5.1]), one has that C(G * ) is also rational.Therefore, one may apply again the theorem to C(G * ).After finitely many applications of the theorem one gets a birational map sending C(G h ) onto a conic (if d is even) or a line (if d is odd) with the same ground field.Finally, taking into account that a line can be parametrized over the ground field, and a conic over a field extension of the ground field of degree at most two, one concludes that there always exists a field of parametrization of C(G h ) that involves a field extension of K of degree at most two (see Corollary 5.9. in [29] for further details).Indeed, this field extension, of degree at most two, is the field extension used in Step (4) of the parametrization computation (see Subsection A.4 in the appendix).
Let us now focus on the case where the ground field is L ∶= K(a); see notation above.Let G ∈ L[x, y] be an irreducible (over L) non-constant polynomial, and let C(G) be rational.We analyze the fields of parametrization of C(G).We observe that if the degree two field extension is L(α), with minimal polynomial t 2 + b t + c ∈ L[t], then L(α) = L(β) where β = α + b/2 has the minimal polynomial t 2 + c − b 2 /4.Therefore, taking into account the above discussion on Hilbert-Hurwitz Theorem, the following holds (see also [29,Corollary 5.9]).
(1) If deg(C(G)), is odd then L is a field of parametrization.
(2) If deg(C(G)) is even then either L is a field of parametrization or there exists δ ∈ L algebraic over L, with minimal polynomial Remark 2.2.Observe that the previous result is valid taking L as any field extension of K.
The case where a contains a single element admits a particular treatment because of Tsen's Theorem; we refer to [5] for this topic.
Proof.By Hilbert-Hurwitz Theorem (see e.g.[29,Theorem 5.8] and as explained above), C(G) is L-birationally equivalent to either a line or a conic.In the line case, the result is clear.In the conic case, since #(a) = 1, the result follows from Tsen's Theorem (see e.g.[31,Corollary 1.11]).□ Remark 2.4.The proof of Tsen's Theorem provides a method for computing a regular point on the conic with coordinates in L.An alternative approach for computing this point can be found in [12] and [32].
Remark 2.5.In the following section we work with G ∈ K[a, γ][x, y] where γ is algebraic over K(a).In the case where #(a) = 1, we can view γ as the only parameter and write a in terms of γ.More precisely, let M (a, c) ∈ K(a)[c] be the minimal polynomial of γ.We can view M as a rational expression in a and consider its numerator as polynomial in b with the root a.Thus, K(γ, a) ∶ K(γ) is a field extension of degree d ≤ deg(H).If d = 1, by Corollary 2.3, K(γ) is a field of parametrization of C(G).
Remark 2.6.In Corollary 2.3, we have seen that if #(a) = 1 then L is a field of parametrization.The following example shows that if #(a) > 1, in general, L is not a field of parametrization.We consider the conic defined by G ∶= a 1 x 2 +a 2 y 2 −1.We prove that C(G) does not have a parametrization over C(a 1 , a 2 ).Let us assume that C(a 1 , a 2 ) is a field of parametrization of C(G), then C(G) has infinitely many points in C(a 1 , a 2 ) 2 .C(G) parametrizes properly as ) , which inverse is So, there are infinitely many points in C(G) ∩ C(a 1 , a 2 ) 2 that are injectively reachable, via P, for t ∈ C( √ a 1 , √ a 2 ).Indeed, note that all points of C(F ), with the exception of (− √ a 1 , 0), are reachable by P. Let t 0 ∈ C( √ a 1 , √ a 2 ) ∖ {0, ±i} be one of these parameter values; say holds that √ a 1 + x 0 , √ a 1 − x 0 are coprime (seen as polynomials in √ a 1 ) and a contradiction.For x 0 = 0 the curve-points (x 0 , y 0 ) = (0, ±1/ √ a 2 ) are not in C(a 1 , a 2 ).

Specializations
Throughout the paper, we will specialize the tuple of parameters a taking values in S (see (1.2)).We will write a 0 to emphasize that the parameters in a have been substituted by elements in K.In the following we discuss different aspects on the specializations.
3.1.General statements.The elements in K(a) are assumed to be represented in reduced form; that is, the numerator and denominator are assumed to be coprime.Then, for f ∶= p(a)/q(a) ∈ K(a), where by assumption gcd(p, q) = 1, and for a 0 ∈ S (see (1.2)) such that q(a 0 ) ≠ 0, we denote by f (a 0 ) the K-element p(a 0 )/q(a 0 ).
We may need to work in the finite field extension F = L(γ) = K(a)(γ).Let p(a, t) ∈ K(a)[t], of degree k in t, be the minimal polynomial of γ.We might simply write p(t) instead of p(a, t) and express it as Then, for a 0 ∈ S such that all D i (a 0 ) ≠ 0, we denote by γ 0 the algebraic element, over K(a 0 ), defined by an irreducible factor of The below reasonings are valid, independently of the irreducible factor taken to define γ 0 .Then, for an element f ∈ F, specialized at a 0 ∈ S, we might simply write f (a 0 ) instead of f (a 0 , γ 0 ).Clearly for a 0 ∈ Ω γ , γ 0 is well-defined.The elements in F are assumed to be expressed in canonical form; that is, f ∈ F is expressed as where U i , W ∈ K[a] and gcd(U 1 , . . ., U k−1 , W ) = 1.In addition, the coefficients of polynomials in F[v], w.r.t. to the tuple of variables v, are also supposed to be written in canonical form.By abuse of notation, we will also denote by f (a, t) the polynomial in L[t] obtained by replacing in (3.2) the element γ by the variable t.Moreover, for f as in (3.2), we denote by Norm(f where the product is taken over all roots γ i in L of the minimal polynomial of γ, say p(a, t) (see (3.1)).Since p is monic, taking into account the expression of the resultant as the product of the evaluations of one of the polynomials in the roots of the other, we get that, up to sign, Norm(f ) = res t (f (a, t), p(a, t)).In particular, Norm(f Proof.D(a 0 )W (a 0 ) ≠ 0. So γ 0 , f (a 0 ) and Norm(f )(a 0 ) = res t (f (a 0 , t), p(a 0 , t)) are welldefined.Since f (a 0 , γ 0 ) = 0 = p(a 0 , γ 0 ), we obtain Norm(f where v is a tuple of variables.Let S be the set of all non-zero coefficients of H w.r.t.v. Let note that C ∈ S ⊂ F is a rational function in a, γ expressed in canonical form and, hence, its denominator is in K[a].And let note that Norm(numer(C)) ∈ L and, hence its numerator is in Remark 3.4.Throughout the paper, we will define several open subsets of S. All these open subsets will be included in Ω γ (for the corresponding algebraic element γ).So, we observe that γ 0 will always be well-defined.
The next lemma justifies the previous definitions.
) is well-defined, and the result follows from the definition of D(H).
Remark 3.7.In the sequel, unless there is a risk of ambiguity, we will omit in the notation of the gcds, and of the open subset Ω gcd , the polynomial ring where the gcd is taken.
In the remaining part of this section, we might write for the specialization of a given polynomial, or rational function, P at a 0 simply P 0 .Lemma 3.8.Let f 1 , f 2 , g be as in Def.3.6.For i , g 0 preserve the degree in v and, in particular, are also non-zero.This implies that f * 0 i are non-zero too.From (3.3), one has that, up to multiplication by non-zero elements in K, gcd Let us prove that δ ∶= deg v (gcd )) = 0, in which case taking λ(u) as this gcd, the gcd equality in the statement of the lemma would hold.Indeed, let If the polynomials are univariate over a field, Lemma 3.8 can be simplified as follows.
, it holds that, up to multiplication by non-zero elements in K, We generalize these results to several univariate polynomials with coefficients in F.
Remark 3.11.Observe that if r = 2 in Def.3.10, then Def.3.6 and 3.10 coincide.In addition, since f 1 does not depend on Z, there exists µ ∈ F ∖ {0} such that gcd and deg v (g * 0 ) = deg v (g * ).Note that, since g * 0 does not depend on Z, then λ ∈ K ∖ {0}.On the other hand, by Remark 3.11, there exists µ ∈ F ∖ {0} such that g * = µg.Furthermore, µ 0 ∶= µ(a 0 , γ 0 ) is well defined.Moreover, since the leading coefficients of g * and g do no vanish when specialized, then µ 0 ≠ 0. Thus, up to multiplication by non-zero elements in K, it holds that Moreover , f (a 0 , γ 0 , v) ≠ 0 and the equality of the degrees holds.Since a 0 ∈ Ω nonZ(R) , also by Lemma 3.5, R(a 0 , γ 0 , v) is well-defined and non-zero.Since A(a 0 , γ 0 , v) ≠ 0, by [38,Lemma 4.3.1], the discriminant of f (a 0 , γ 0 , v) is not zero.Then, by [38,Theorem 4.4.1],f (a 0 , γ 0 , v) is squarefree.□ 3.2.Specialization of the curve defining polynomial.In this subsection, we deal with the specialization of defining polynomials of irreducible plane curves.Let G ∈ F[x, y] ∖ F be irreducible over F of total degree d.In the following, let G be written as where g i is either the zero polynomial or a form of degree i.

Specialization of families of points.
Let us now deal with the specialization of conjugate families of points appearing in the standard decomposition of the singular locus of the curve.In Subsections A.1 and A.2 of the appendix, a description of these concepts appears; see in particular Definitions A.1 and A.9 and Remark A.12.
Let G and G h be as in Subsection 3.2.For a 0 ∈ S such that G(a 0 , x, y) / ∈ K, G(a 0 , x, y) defines a plane curve over K. Let C(G) denote the curve defined by G(a, γ, x, y) over F, and C(G, a 0 ) the curve defined by G(a 0 , γ 0 , x, y) over K. Similarly for the projective curves.
Conjugate families of a curve are referred to a field.The conjugate families of C(G h ) will be over F. When we specialize a we need to have a reference field where the conjugation of the points is defined.This motivates the following definition.Definition 3.17.For a 0 ∈ S, we define K a 0 as the smallest subfield of K containing the coefficients of G(a 0 , γ 0 , x, y).
Let D(G h ) denote the F-standard decomposition of the singular locus of C(G h ) obtained using the process described in Subsection A.2; see also Def.A.9 and Remark A.12.
We start our analysis with a technical lemma.In the remaining part of this section, we might write for the specialization of a given polynomial, or rational function, P at a 0 simply P 0 .
Let R be the remainder of the division of H by m w.r.t.t, and let A be the leading coefficient of m w.r.t.t.If H(a 0 , t), m(a 0 , t) are well-defined and A(a 0 ) ≠ 0, then H(a 0 , t) = R(a 0 , t) mod m(a 0 , t).
m) (see Def. 3.10) then γ 0 , f 0 i,m and m 0 are well-defined and, by Theorem 3.12, the gcd(f 0 1,m , f 0 2,m , m 0 ) = 1.Furthermore, since a 0 ∈ Ω NonZ(A) (see Def. 3.3), the degree of f i,m and m is preserved under the specialization.In addition, since a 0 ∈ Ω sqfree(m) , it holds that m 0 is squarefree (see Lemma 3.14).So the conditions in Def.A.1 hold.Furthermore, note that, after specialization, all polynomials are over K a 0 .So, F(a 0 ) is a family over K a 0 .It remains to prove that the points in F(a 0 ) are in the specialized curve.Since a 0 ∈ Ω G , by Lemma 3.16, it holds that are well-defined, then T 0 is well-defined too.We know that m 0 is well-defined and that the leading coefficient of m in t does not vanish after specialization.Therefore, by Lemma 3.19, all the arguments above apply and, hence, one deduces that F(a 0 ) is a K a 0 -family of points of C(G h , a 0 ).
It remains to prove that #(F(a 0 )) = #(F).Let L be the K-linear change of coordinates transforming C(G) in regular position; see Step (1) in the standard decomposition process described in Subsection A.2.Then, #(F) = #(L −1 (F)).Furthermore, L −1 (F) is in the form appearing either in (A.1) or in (A.2).Therefore, #(F) = deg t (m).Since L is over K, we may apply it to F(a 0 ) and L −1 (F(a 0 )) will be of the form either In both cases, #(F(a 0 )) = #(L −1 (F(a 0 ))) = deg t (m 0 ).Now, the result follows using that deg t (m) = deg t (m 0 ).□ Remark 3.21.Given an F-family F ∈ D(G h ) (see (3.5)), and a 0 ∈ Ω def(F ) (see Def. 3.18), we observe that, even though F is irreducible, F(a 0 ) may be reducible.We are interested in working with irreducible specialized families.So, factoring over K a 0 the defining polynomial of F(a 0 ), the family will be decomposed as where F i is an irreducible K a 0 -family.We refer to F i as the irreducible subfamilies of F(a 0 ).Note that, if In the sequel we analyze the multiplicity of families of singularities under specializations.If a 0 ∈ Ω mult(F ) , then every irreducible subfamily of F(a 0 ) (see Remark 3.21) is a K a 0 -family of r-fold points of C(G h , a 0 ).
Proof.F can be expressed as , m irreducible over F, and such that, for the case λ = 0, deg t (f i ) ≤ 1 and gcd(f 1 , f 2 ) = 1 (see (3.5)).Let H * , H be as in Def.3.22.Since a 0 ∈ Ω def(F ) , by Lemma 3.20, F(a 0 ) is a K a 0 -family of points of C(G h , a 0 ).Let F i ∶= {(f 1 (a 0 , t) ∶ f 2 (a 0 , t) ∶ λ)} m * be an irreducible subfamily of F(a 0 ).Now, let W (a, x, y, z) be any partial derivative of G h of order smaller than r.
Then, W (a, f 1 (a, t), f 2 (a, t), λ) = 0 mod m(a, t).Thus, since all the involved specializations are well-defined, W (a 0 , f 1 (a 0 , t), f 2 (a 0 , t), λ) = 0 mod m * .Therefore, the points at F i have multiplicity at least r.On the other hand, since a 0 ∈ Ω G , H * (a 0 , f 1 (a 0 , t), f 2 (a 0 , t), λ) is welldefined and deg(m(a, t)) = deg(m(a 0 , t)), by Lemma 3.19, H * (a 0 , f 1 (a 0 , t), f 2 (a 0 , t), λ) = H(a 0 , t) modulo m(a 0 , t).Furthermore, since a 0 ∈ Ω nonZ(H) , then H(a 0 , t) ≠ 0.Moreover, since the leading coefficient of m w.r.t.t does not vanish at a 0 , we have that res t (H(a 0 , t), m(a 0 , t)) = µ R(a 0 ) for some non-zero constant µ ∈ K (see Lemma 4.3.1 in [38]).So, since a 0 ∈ Ω nonZ(R) , res t (H(a 0 , t), m(a 0 , t)) ≠ 0. Therefore, gcd(H(a 0 , t), m(a 0 , t)) = 1 and hence H(a 0 , t) ≠ 0 mod m * .Summarizing, mult(F i ) = r.□ In the last part of this subsection, we deal with the tangents to C(G h ) at an irreducible F-family.For this purpose, since the family F is irreducible, we will work with the curve C F (G h ) associated to F (see Def. A.4 in Subsection A.1 of the appendix).Definition 3.25.Let F = {(f 1 ∶ f 2 ∶ f 3 )} m be an irreducible F-family of r-fold points of C(G h ).Let F m be the quotient field of F[t]/ < m(t) >.The defining tangent polynomial of C(G h ) at F is the homogenous polynomial T ∈ F m [x, y, z] of degree r that defines the tangents, with its corresponding multiplicities, to C F (G h ) at the point (f 1 ∶ f 2 ∶ f 3 ).Similarly, we introduce the defining tangent polynomial to a specialized curve.
Remark 3.26. ( )) be an irreducible family of affine r-fold points; similarly if the family is at infinity.The defining tangent polynomial T is the reduction of (2) Let F be as in Def.3.25.Then F is a family of ordinary points if and only if T is squarefree over F m .In the sequel, we assume w.l.o.g. that there is no tangent to C F (G h ) at F independent of x and that for two different tangents T 1 (a, x, y, z), T 2 (a, x, y, z) it holds that T 1 (a, x, 1, 1) ≠ T 2 (a, x, 1, 1).Note that, if this is not the case, one can apply a linear change over K (and thus invariant under specializations of the parameters a).Then, the ordinary character of the family is readable from the squarefreeness of T (a, t, x, 1, 1) over F m .
Definition 3.27.Let F, with defining polynomial m(t), be an irreducible F-family of ordinary r-fold points of C(G h ).Let T be the defining tangent polynomial of F, where we assume w.l.o.g. that the hypotheses in Remark 3.26 (2) are satisfied.Let D(a, t) be the reduction modulo m(t) of the discriminant w.r.t.x of T (a, t, x, 1, 1).Let N (a) = res t (D, m).
Let A(a, t) be the leading coefficient of T (a, t, x, 1, 1) w.r.t.x and let R(a) = res t (A, m).Let S(a, x) = res t (T (a, t, x, 0, 0), m).We define the set Remark 3. (3) T (a, t, x, y, z) has a factor in F m [y, z] if and only if T (a, t, x, 0, 0) = 0.This follows from the fact that the tangents are of degree one and thus, T (a, t, x, y, z) = ∏(Ai(a, t)x + B i (a, t)y + C i (a, t)z) for some A i , B i , C i ∈ F m .Thus, under our assumption that T (a, t, x, y, z) does not have a factor independent of x, T (a, t, x, 0, 0) ≠ 0. Since m is irreducible, and deg t (T (a, t, x, 0, 0)) < deg t (m), it follows that S ≠ 0.
)) be an irreducible F-family of ordinary r-fold points of C(G h ).If a 0 ∈ Ω ord(F ) , every irreducible subfamily of F(a 0 ) (see Remark 3.21) is a K a 0family of ordinary r-fold points of C(G h , a 0 ).
) and, by Lemma 3.24, every irreducible subfamily of F(a 0 ) is a K a 0 -family of r-fold points of C(G h , a 0 ).Let us prove that all points in F(a 0 ) are ordinary.Let T * , T be as in Remark 3.26 (1), or similarly if the family is at infinity.Since a 0 ∈ Ω ord(F ) ⊂ Ω mult(F ) ⊂ Ω def(F ) ⊂ Ω G , by Lemma 3.16, T * specializes properly at a 0 , and since deg t (m(a, t)) = deg t (m(a 0 , t)), by Lemma 3.19, T also specializes properly at a 0 .Now, let P be a point in F(a 0 ).Then, there exists a root t 0 of m(a 0 , t) such that P is obtained by specializing F at a 0 and t 0 .Since P belongs to one of the irreducible subfamilies, P is an r-fold point of the curve C(G h , a 0 ).Because of the discussion above, E(x, y, z) ∶= T (a 0 , t 0 , x, y, z) is the defining tangent polynomial of C(G h , a 0 ) at P .It remains to prove that E is squareefree.First, let us see that there is no factor of E independent of x.Assume that e(y, z) is a factor of E. Then E(x, 0, 0) = 0, and (a 0 , t 0 , x, 0, 0) is a common zero of T (a, t, x, y, z) and m(a, t).Therefore, see e.g.Theorem 4.3.3 in [38], S(a 0 , x) = 0 which contradicts that a 0 ∈ Ω nonZ(S) .Thus, it is sufficient to prove the squarefreeness of E(x, 1, 1).Let us assume that it is not squarefree, then its discriminant is zero.That is, the discriminant of T (a 0 , t 0 , x, 1, 1) is zero.On the other hand, a 0 ∈ Ω nonZ(R) , R(a 0 ) ≠ 0 and thus, A(a 0 , t 0 ) ≠ 0. By [38,Lemma 4.1.3]and the fact that m(a 0 , t 0 ) = 0, it follows that D(a 0 , t 0 ) = 0 and consequently, N (a 0 ) = 0, in contradiction to a 0 ∈ Ω nonZ(N ) .□ As a consequence of Lemmas 3.20, 3.24, 3.29, and taking into account that Ω ord(F ) ⊂ Ω mult(F ) ⊂ Ω def(F ) , we get the following corollary.

Preservation of the Genus
We consider a polynomial (4.1) F , as a non-constant polymomial in F[x, y], defines an affine plane curve over F that we assume irreducible.As introduced in Subsection 3.3, for each a 0 ∈ S such that F (a 0 , γ 0 , x, y) / ∈ K, we denote by C(F, a 0 ) the curve C(F (a 0 , γ 0 , x, y)); similarly for C(F h , a 0 ).Also, we denote by K a 0 the ground field of C(F, a 0 ) (see Def. 3.17).Our goal is to analyze the relation between the genus of C(F ) and the genus of C(F, a 0 ), under the assumption that C(F, a 0 ) is irreducible.
4.1.Ordinary singular locus case.We start our analysis assuming that C(F h ) has only ordinary singularities.Let D(F h ) be the F-standard decomposition of the singular locus of C(F h ) obtained by using the process described in Subsection A.1.Let D(F h ) decompose as in (3.5) Then, the following result holds.If sing(C(F h )) ≠ ∅, then a 0 ∈ Ω singOrd(F h ) ⊂ Ω F and deg(F (a 0 , γ 0 , x, y)) = d.By Corollary 3.30, all elements in sing(C(F h )) have the same multiplicity and character as their corresponding elements in sing(C(F h , a 0 )) after specialization.New singularities, however, may appear in sing(C(F h , a 0 )).So, reasoning as above with the genus formula in (A.4), or (A.8), in the appendix, we get the result.□ The next result is a direct consequence of the previous theorem when the genus of C(F ) is zero.
The inequality in Theorem 4.2 comes from the fact that, using Ω SingOrd(F h ) , we cannot ensure that sing(C(F h , a 0 )) does not include new singularities apart from those coming from the specialization of the singular locus of C(F h ).To control this phenomenon, we will ensure that certain Gröbner bases behave properly under specializations.By, exercises 7, 8, pages 315-316 in [4], or by Proposition 1, page 308 in [4], we know that there exists an open Zariski set such that the Gröbner basis specializes properly; in fact, a description of this open subset is also available.For a more general analysis of Gröbner bases with parametric coefficients we refer to [17] and [37].On the other hand, since we are working with bivariate polynomials in F[x, y], the open subset above can be determined by using resultants.This motivates the next definition.Let G be a Gröbner basis of G w.r.t.some order.We define Ω spGB(G) ⊂ S as a non-empty open subset such that for every a 0 ∈ Ω spGB(G) it holds that {g(a 0 , γ 0 , v) | g ∈ G} is a Gröbner basis, w.r.t. the same order, of the ideal generated by {g(a 0 , γ 0 , v) Now, we focus our attention on the standard decomposition of the singular locus of C(F h ) described in (4.2).In the first step, if necessary, we apply a K linear change of coordinates to ensure that the curve is in regular position.Hence, this linear transformation it is not affected by the specializations of a.Therefore, for our reasonings, we may assume w.l.o.g. that F is already in regular position.Next, let G 1 be a Gröbner basis of <F, F x , F y > w.r.t. the lexicographic order with x < y, and let G 2 be a Gröbner basis of the same ideal w.r.t. the lexicographic order with y < and g = g/ gcd(g, g y ), where f x is the derivative of f w.r.t.x; similarly with g y .Finally, let G 3 ∶= {A(a, x), y − B(a, x)}, with A square-free and deg(B) < deg(A), be the normed reduced Gröbner basis w.r.t. the lexicographic order with x < y of <F, F x , F y , f , g>.Then, we introduce the following definitions (recall Def.3.3, 3.6, 3.13, 3.10, 3.13 and 4.1).Definition 4.5.(Affine singularities) With the notation introduced above, let where U and V are the leading coefficients of f and g w.r.t.x and y, respectively.We define the open subset where W q,y denotes the leading coefficient of q w.r.t.y; similarly with W q,x .Remark 4.6.Note that all polynomials in G 1 ∖{f } do depend on y; similarly for G 2 ∖{g}.The idea of controlling the coefficients W q,x and W q,y in Def.4.5 is to ensure that the elimination ideal of the specialized Gröbner basis does not include additional generators.
Proof.Let us denote the specialization of a polynomial Q at a 0 as Q 0 .Let G 0 1 be a Gröbner basis of <F 0 , F 0 x , F 0 y > w.r.t. the lexicographic order x < y, and let G 0 2 be a Gröbner basis of the same ideal w.r.t. the lexicographic order y < x.Since .
By Lemma 3.14, it holds that deg x ( f ) = deg x ( f 0 ) and f 0 is squarefree.Similarly for g and g since a 0 ∈ Ω 2 .In addition, by Lemma 3.16, , by Lemma 3.14, A 0 is squarefree.Therefore, the number of affine singularities of C(F, a 0 ) is deg x (A 0 ) and deg x (A) is the number of affine singularities of C(F ).By Lemma 3.14, we get that deg x (A 0 ) = deg x (A) and, hence, C(F ) and C(F, a 0 ) have the same number of affine singularities.
Let D(a, γ, t) = gcd(U, U ′ ) and D 0 (t) = gcd(U 0 , (U 0 ) ′ ).Since a 0 ∈ Ω gcd(U,U ′ ) , by Corollary 3.9, one has that Proof.Since a 0 ∈ Ω sing a (F ) ∩ Ω sing ∞ (F ) , by Lemma 4.9, it holds that #(sing(C(F h )) = #(sing(C(F h , a 0 )).On the other hand, since a 0 ∈ Ω SingOrd(F h ) , by Corollary 3.30, we know that each ordinary r-fold in sing(C(F h )) generates an ordinary r-fold in sing(C(F h , a 0 )).Therefore, applying the genus formula (A.4) in the appendix, we conclude the proof.□ 4.2.General case.Let F be as in (4.1).But now, differently to the case of Subsection 4.1, we do not introduce any assumption on the singular locus of the irreducible curve C(F h ).The key of our analysis is to reduce the general case to the case studied in Subsection 4.1.For this purpose, we recall that any irreducible curve is birationally equivalent to a curve having only ordinary singularities; see e.g.[36,Theorem 7.4.],[3, Theorem 9.2.4] or [29, Section 3.2.]for a more computational description.This transformation, say φ, can be seen as a finite sequence of blowups of the irreducible families of non-ordinary singularities and, hence, as a finite sequence of compositions of quadratic Cremona transformations and linear transformations.We conclude by the fact that the genus is invariant under birational transformations.Now, our goal is to find an open subset Ω blowup of S such that, when a is specialized in Ω blowup , the birationality of φ is preserved.For this purpose, let F 0 (x, y) = F (x, y) and let F(F h 0 ) be an irreducible F-family of non-ordinary singularities of Then we apply a linear transformation L 1 , given by a matrix M 1 ∈ M 3×3 (F m 1 ), and the Cremona transformation Q 1 = (yz ∶ xz ∶ xy) as described in the blow up basic step in Subsection A.3 of the appendix.Let ∆ 1 ∶= det(M 1 ) and let C(F h 1 ) be the curve over F m 1 obtained after the quadratic transformation ) not being divisible by neither x, y nor z.We repeat the above process for F h 1 (t 1 , x, y, z), F h 2 (t 1 , t 2 , x, y, z), . . ., F h r (t 1 , . . ., t r , x, y, z) until all singularities of C(F h r ) are ordinary.Then φ(a, γ, t 1 , . . ., t r , x, y, z) Note that F h r is defined over F(t 1 , . . ., t r ) and C(F h r ) over the algebraic closure of F(t 1 , . . ., t r ).In addition, F(t 1 , . . ., t r ) = K(a, γ, t 1 , . . ., t r ) = L(γ, t 1 , . . ., t r ).So, we consider a primitive element of the extension over L, say γ * , and we work over L(γ * ) = L(γ, t 1 , . . ., t r ); note that results in Section 3 apply to this new frame.In this situation, let us denote by ∆ = ∆ 1 ⋯∆ r ∈ L(γ * ) the product of the determinants of the linear transformations L 1 , . . ., L m .In addition, let M ∶= {all entries of and let ))(a, γ * , x, y, 0)} i∈{0,...,r−1} .Definition 4.11.With the notation introduced above, we define the set The previous observations lead to the following result.Lemma 4.12.Let a 0 ∈ Ω blowup (F ).Then φ(a 0 , (γ * ) 0 , x, y, z) is birational.
Proof.First we observe that because of (the proof of) Lemma 4.12 each φ i ∶= Q i ○ L i is well defined at a 0 and it is birational.Let us prove the result by induction.By hypothesis F h 0 (a 0 , γ 0 , x, y, z) is irreducible.We have the equality for some n i ∈ N and that neither x, y nor z divides F h 1 .So, F 0 (φ i )(a 0 , (γ * ) 0 , x, y, z) = x n 1 y n 2 z n 3 F h 1 (a 0 , (γ * ) 0 , x, y, z).Moreover, since a 0 ∈ ⋂ h∈B Ω nonZ(h) , we know that neither x, y nor z divides F 1 (a 0 , (γ * ) 0 , x, y, z).Furthermore, since φ i is birational when specialized at a 0 and F h 0 (a 0 , γ 0 , x, y, z) is irreducible, we have that F h 1 (a 0 , (γ * ) 0 , x, y, z) is also irreducible.Thus, F h 1 (a 0 , (γ * ) 0 , x, y, z) is the quadratic transformation of F h 0 .Now, the i-induction step is reasoned analogously using that, by induction, F h i (a 0 , (γ * ) 0 , x, y, z) is irreducible.□ With this, we can now give an open set where the genus is preserved.
Definition 4.14.Let F ∈ F[x, y] be as in (4.1), and let G ∈ F(γ * )[x, y] be the polynomial obtained after the blowup process of F h .We define the set Theorem 4.15.Let F ∈ F[x, y] be as in (4.1), and let Proof.We start recalling that the quadratic transformation defines a birational map (see e.g.[36, Chapter V, Section 4.2]) from the original curve on its quadratic transformed curve (see the precise definition of transformed curve in Step 2 of the blow up at a point in Subsection A.3 in the appendix), and that the genus is a birational invariant (see e.g.[3, Theorem 7.2.2 or Prop.8.5.1]).Then, since G h is the quadratic transformation of F h , we have that Let φ 0 denote the map φ(a 0 , (γ * ) 0 , x, y, z).Since a 0 ∈ Ω blowup(F ) , by Lemma 4.12, φ 0 is birational and, by Lemma 4.13, G h (a 0 , (γ * ) 0 , x, y, z) is the quadratic transformation of F h (a 0 , γ 0 , x, y, z) via φ 0 .Therefore, using the same argumentation as in (4.6), on gets Moreover, since a 0 ∈ Ω genusOrd(G h ) , by Theorem 4.10, it holds that Now, the proof follows from (4.6), (4.7) and (4.8).□

Birational Parametrization of Parametric Rational Curves
In Section 4, and more precisely in Theorems 4.10 and 4.15, we have described open subsets of S where the genus of the curve is preserved under specializations; even in Corollary 4.3 the particular case of genus zero was treated.Nevertheless, in all these results the additional condition on the irreducibility, over K, of the specialized polynomial was required.Avoiding the irreducibility is in general a difficult problem related to the Hilbert irreducibility problem (see e.g.[30]).More precisely, there is no algorithm known that finds for given irreducible F ∈ F[x, y] the specializations a 0 ∈ S such that F (a 0 , x, y) is reducible.Nevertheless, in this section, we will analyze the particular case where F defines a genus zero curve.In this case, we will introduce an open subset where the irreducibility is guaranteed.
For this purpose, throughout this section, let assume that F ∈ F[x, y] is as in (4.1) and additionally assume that C(F ) is rational.Moreover, let us assume that P is a proper (i.e.birational) parametrization of C(F ) which can be computed, for instance, by the algorithm described in Subsection A.4 in the appendix.Note that, in general, one may need to extend F with an algebraic element δ of degree two (see Subsection 2.2).If #(a) = 1 and deg(γ) = 1, or deg x,y (F ) is odd, then no extension of F is required (see Remark 2.5 and Theorem 2.1).
So, we may consider a primitive element of L(γ, δ), say γ * , and express our parametrization in L(γ * ).Throughout this section, by abuse of notation, let γ play the role of γ * and let F denote the field L(γ * ).Let us write the proper parametrization P of C(F ) as (5.1) where we assume that P is in reduced form, that is gcd(p 1 , q 1 ) = gcd(p 2 , q 2 ) = 1.
Let us start with the simple case of degree one.
is a line, it can be properly parametrized by a polynomial parametrization of the form P(a, γ, t) = (λ We define the set (see Def. 3.3 and 3.15) Proposition 5.2.Let F and P be as in Def.5.1.For every a 0 ∈ Ω proper(P) , it holds that P(a 0 , t) is a proper polynomial parametrization of C(F, a 0 ).
In the sequel, we assume that C(F ) is not a line.We generalize the open subset in Def.5.1 as follows.
Finally, since H(x, y) divides F 0 , one has that C(F, a 0 ) = C(H), which concludes the proof.□ Remark 5.6.Let us analyze the behavior of F and/or P when specializing in S ∖ Ω proper(P) .
(2) If a 0 ∈ S ∖ Ω 2 , then P(a 0 , t) is not defined, and hence the specialization fails. ( , at least one of the following assertions hold.(a) P(a 0 , t) is not proper.(b) P(a 0 , t) ∈ K 2 and hence, P(a 0 , t) is not a parametrization.
(c) P(a 0 , t) parametrizes a proper factor of F (a 0 , x, y), that is, C(F, a 0 ) decomposes and one of its components is rational and parametrized by P(a 0 , t).
The next result follows from Theorem 5.5 and emphasizes the polynomiality of the parametrizaion.
We now analyze the normality (i.e. the surjectivity, see [22], [29]) of the parametrization.We recall that any parametrization can be reparametrized surjectively (see [29,Theorem 6.26]).This reparametrization requires, in our case, a new algebraic extension of F via a new algebraic element.Alternatively, one may reparametrize normally the specialized parametrizations.In the following we deal with the case where P is already normal and we want to preserve this property through the specializations.For this purpose, we first introduce a new definition.We recall that for an affine plane parametrization P of a curve C, C ∖ P(K) contains at most one point.That point, if it exists, is called the critical point of the parametrization (see [29,Def. 6.24]).Definition 5.8.Let P be as in (5.1).If P is normal, we define the set where (α 1 /β 1 , α 2 /β 2 ) ∈ F 2 is the critical point of P and, for i ∈ {1, 2}, N i = α i q i − β i p i , Corollary 5.9.Let P be proper and normal.For a 0 ∈ Ω proper(P) ∩ Ω normal(P) , P(a 0 , γ 0 , t) parametrizes properly and normally C(F, a 0 ).
Proof.As in the proof of Theorem 5.5, we use the same criterion for simplifying the notation.Namely, if Q is a polynomial, or a rational function, we will denote by Q 0 its specialization at a 0 .Similarly P 0 will denote the specialization at a 0 of the parametrization P. In the proof of Theorem 5.5 we have seen that, for a 0 ∈ Ω proper(P) , deg t (p i ) = deg t (p 0 i ), similarly for q i , and that the rational functions in P 0 are in reduced form.Now, if deg t (p 1 ) > deg t (q 1 ) or deg t (p 2 ) > deg t (q 2 ), the result follows from Theorem 5.5 and [29,Theorem 6.22].If deg t (p 1 ) ≤ deg t (q 1 ) and deg t (p 2 ) ≤ deg t (q 2 ), since a 0 ∈ Ω 2 in Def.5.4, we get that is well defined and, by the above remark on the degrees, C is the critical point of P 0 .Now, since a 0 ∈ Ω gcd(N 1 ,N 2 ) , by Corollary 3.9, one has that deg t (gcd(N 0 1 , N 0 2 )) = deg t (gcd(N 1 , N 2 )) > 0; recall that P is normal.Now, the result follows from Theorem 5.5 and [29,Theorem 6.22].□ Let us illustrate these ideas in an example.
The field of parametrization is L. We determine the open subset Ω proper(P) (see Def. (5.4)).
Let us deal with Ω 1 .Clearly Ω def(F ) = C 2 .The homogeneous component of F of maximum degree is One has that Ω 2 ∶= C 2 ∖ {(0, 0)}.Note that, if a 0 / ∈ Ω 2 , the second component of P is not well-defined.Let us deal with Ω 3 .The polynomials G i , and Ω nonZ(B) = C 2 .On the other hand, Ω nonZ(R) can be expressed as Therefore, Finally, we deal with Ω 4 .We have The goal in this section is to provide an algorithm decomposing the space S so that in each subset of the decomposition we can give information on the genus of the corresponding specialized curve.
Let F be as in (4.1), irreducible over F. We first compute the genus of C(F h ).Let g ∶= genus(C(F h )).Furthermore, if g = 0, let P(a, γ, t) be, as in (5.1), a proper parametrization of C(F h ).We consider the open subset At this level of the process we know that (see Theorems (4.15) and (5.5)) (1) If g > 0, then for a 0 ∈ Σ it holds that C(F, a 0 ) is either reducible or its genus is g.
(2) If g = 0, then for a 0 ∈ Σ it holds that C(F, a 0 ) is rational and P(a 0 , γ 0 , t) parametrizes properly C(F, a 0 ).In the following, we analyze the specializations when working in the closed set Finally, since Σ of F as in (6.1) is open and non-empty, depending on the genus of F , either S 2 or S 3 is a dense subset of S.
We illustrate the previous ideas by continuing the analysis of Example 5.10.
We start with J 1 ∶=<a 1 > and V 1 ∶= V(J 1 ).Since V 1 is rational, surjectively parametrized by Q 1 ∶= (0, λ), we work over the field Q(λ)[x, y].We have that and therefore all specializations in V(a 1 ) lead to a reducible curve.Additionally, one may distinguish the cases λ = 0, that corresponds to the point (0, 0), where the specialization degenerates, and λ ≠ 0 where C(F, a 0 ) decomposes to the union of a double line and a rational cubic.The analysis for J 2 ∶=<a 2 >, and V 2 ∶= V(J 2 ) looks similar.Since V 2 is rational, parametrized by Q 2 ∶= (λ, 0), we work over the field Q(λ)[x, y].We have that Thus, all specializations in V 2 lead to a reducible curve; note that Q 2 is surjective.The case λ = 0 is covered above, and for λ ≠ 0, the specialization C(F, a 0 ) decomposes to the union of a line and a rational quartic.
For a 0 ∈ S 2 , C(F, a 0 ) is either reducible or an elliptic curve; and for a 0 ∈ S 3 , C(F, a 0 ) is a rational cubic parametrized by P J 3 (a 0 , t).

Some Illustrating Applications
In this section, we illustrate by examples some possible applications of the theory developed in the paper.In the first example, given a surface, we consider the problem of determining its rational level curves, if any.Using Theorem 4.15, for z 0 ∈ Ω genusOrd(F ) , C(F, (x, y, z 0 )) is either reducible or its genus is 9.In any case, no rational level curve appears.For the elements in Z ∶= C ∖ Ω genusOrd(F ) , we get that C(F, (x, y, ±1)) are irreducible of genus 7 and C(F, (x, y, 0)) is irreducible of genus 0. Indeed, C(F, (x, y, 0)) can be parametrized by (t 5 , t 6 − t 2 ).
In the second example, we consider the linear homotopy deformation of two curves and we analyze the genus of each instance curve.Let O be a connected open subset of C and let Mer(O) be the field of meromorphic functions in O (see [14]).We consider a polynomial equation of the form (7.1) ∑ i,j∈I f i,j (t)x i y j = 0 where I is a finite subset of N 2 , and where f i,j ∈ Mer(O).Let f be the tuple with all the functions f i,j appearing in (7.1).The question now is to decide, and indeed compute, whether there exists rational solutions of the equation; that is p, q ∈ C(f ) such that ∑ i,j∈I f i,j (t)p i q j = 0. We may proceed as follows.We consider the polynomial F (a, x, y) resulting from the formal replacement in (7.1) of each function f i,j by a parameter a k .Now, with the terminology of the paper, we take K = C(f ) and L = F = K(a).
Then, we decompose S (see (1.2)) as described in Section 6; note that the computations can be carried out over C(a) instead of over F.Then, if any subset in the decomposition has genus zero, and the functions f belong to it, we obtain a (family of) rational solutions.Let us see a particular example.
3.6 and Lemma 3.5 (1)).So,(3.3) and where A a and A ∞ are finite sets of irreducible polynomials in F[t].By abuse of notation, we will write F ∈ D(G h ) to refer to the families in D(G h ).Definition 3.18.Let F ∶= {(f 1,m ∶ f 2,m ∶ 1)} m ∈ D(G h ) be an irreducible F-family of affine singularities of C(G h ) (see (3.5)).Let A be the product of the leading coefficients w.r.t.t of m, f 1,m , f 2,m .We associate to F the open set

28 .
In relation to Def. 3.27, we observe the following.(1) By construction, deg t (A) < deg t (m) and clearly A is not zero.Since m is irreducible, gcd(A, m) = 1.Hence, R ≠ 0. (2) Since F is ordinary and the two hypotheses in Remark 3.26 (2) are satisfied, D ≠ 0. Because deg t (D) < deg t (m) and m is irreducible, gcd(m, H) = 1 and hence, N ≠ 0.

Definition 4 . 4 .
Let I be an ideal in F[v], where v is tuple of variables, generated by G ⊂ F[v].

(6. 2 )
Z ∶= S ∖ Σ.First, let us discuss the computational issues that may appear.Let A ⊂ K[a] be a set of generators of Z, and let I be the ideal generated by A in K[a].We consider the prime decomposition of I I = ℓ ⋃ j=1 I j .

Figure 2 .
Figure 2. Left: Plot of the real part of different instances of the deformation in Example 7.2.Right: Plot of the real part of C(F (3/4, x, y)) and C(F, (3, x, y)) in Example 7.2.
[29,rationality of C(G) and C(G h ) are equivalent.Moreover, parametrizations of C(G) and C(G h ) relate each other by means of homogenizing and dehomogenizing.So, in the following we focus on affine parametrizations.A parametrization P(t) is called birational or proper if the map K ⇢ C(G); t ↦ P(t) is injective in a non-empty open Zariski subset of K (see e.g.[29,Sections 4.2.and4.4]).
, by definition of the open set, the leading coefficient of g w.r.t.v does not vanish at a 0 .Therefore, deg v (g 0 ) = deg v (g).□Our next step is to analyze the squarefreeness.
Definition 3.13.Let f ∈ F[v] ∖ F be squarefree.Let R be the discriminant of f w.r.t.v and let A be the leading coefficient of f w.r.t.v.We define the open subset )) be an irreducible Ffamily of r-fold points of C(G h ) (see Def. A.5), and let H * (a, x, y, z) be one of the order r derivatives of G h such that H * (a, f 1 , f 2 , f 3 ) ≠ 0 modulo m(a, t) (see Remark A.6).Let H(a, t) be the remainder of the division of H * (a, f 1 , f 2 , f 3 ) by m(a, t) w.r.t.t.Let R(a) ∶= res Lemma 3.24.Let F ∈ D(G h ) (see (3.5)) be an irreducible F-family of r-fold points of C(G h ).