Resultants and Singularities of Parametric Curves
aa r X i v : . [ m a t h . AG ] J u l Resultants and Singularities of Parametric Curves
Angel Blasco and Sonia P´erez–D´ıazDepartamento de F´ısica y Matem´aticasUniversidad de Alcal´a28871-Alcal´a de Henares, Madrid, [email protected], [email protected]
Abstract
Let C be an algebraic space curve defined parametrically by P ( t ) ∈ K ( t ) n , n ≥
2. In this paper, we introduce a polynomial, the
T–function , T ( s ), which isdefined by means of a univariate resultant constructed from P ( t ). We showthat T ( s ) = Q ni =1 H P i ( s ) m i − , where H P i ( s ) , i = 1 , . . . , n are polynomials(called the fibre functions ) whose roots are the fibre of the ordinary singular-ities P i ∈ C of multiplicity m i , i = 1 , . . . , n . Thus, a complete classificationof the singularities of a given space curve, via the factorization of a resultant,is obtained. Keywords:
Rational curve parametrization; Singularities of an algebraiccurve; Multiplicity of a point; Tangents; Resultant; T–function; Fibrefunction
1. Introduction
Parametrizations of rational curves play an important role in many prac-tical applications in computer aided geometric design where objects are oftengiven and manipulated parametrically (see e.g. [9], [10], [11]). In the lastyears, important advances have been made concerning the information onemay obtain from a given rational parametrization defining an algebraic vari-ety. For instance, a complete analysis of the asymptotic behavior of a givencurve has been carried out in [2]; efficient algorithms for computing the im-plicit equations that define the curve are provided in [4] and [18] and thestudy and computation of the fibre of a point via the parametrization can befound in [18]. In addition, some aspects concerning the singularities of the
Preprint submitted to Elsevier March 28, 2018 urve and their multiplicities are studied in [1], [5], [6], [13] and [17]. Sim-ilar problems, for the case of a given rational parametric surface, are beinganalyzed. For instance, the computation of the singularities and their mul-tiplicities from the input parametrization is presented in [16], a univariateresultant-based implicitization algorithm for surfaces is provided in [15], andthe computation of the fibre of rational surface parametrizations is developedin [14].In this paper, we show how to relate the fibre and the singularitiesof a given curve defined parametrically, by means of a univariate resul-tant which is constructed directly from the parametrization. For this pur-pose, we consider P ( t ) ∈ P n ( K ( t )) a rational projective parametrizationof an algebraic curve C over an algebraically closed field of characteris-tic zero, K . Associated with P ( t ), we consider the induced rational map ψ P : K −→ C ⊂ P n ( K ); t ( t ) . We denote by deg( ψ P ) the degree ofthe rational map ψ P . The birationality of ψ P , i.e. the properness of P ( t ), ischaracterized by deg( ψ P ) = 1 (see [8] and [19]). Intuitively speaking, P ( t )proper means that P ( t ) traces the curve once, except for at most a finitenumber of points. We will see that, in fact, these points are the singularitiesof C .We recall that the degree of a rational map can be seen as the cardinalityof the fibre of a generic element (see [19]). We use this characterization inour reasoning and thus, we denote by F P ( P ) the fibre of a point P ∈ C viathe parametrization P ( t ); that is F P ( P ) = P − ( P ) = { t ∈ K | P ( t ) = P } . In order to make the paper more reader–friendly, we first consider thecase of a given plane curve C defined parametrically by P ( t ) ∈ P ( K ( t )) (seeSections 2 and 3) to, afterwards, generalize the results obtained to rationalspace curves in any dimension (see Section 4). We also assume that C hasonly ordinary singularities (otherwise, one may apply quadratic transforma-tions for birationally transforming the curve into a curve with only ordinarysingularities). Non–ordinary singularities have to be treated specially sincea non–ordinary singularity might have other singularities in its “neighbor-hood”. This specific case will be addressed in a future work and in fact, wewill show that similar results to those presented in this paper can be statedfor curves with non–ordinary singularities.Under these conditions, the main goal of the paper is to prove that aunivariate resultant constructed directly from P ( t ), which we will call the T–function , T ( t ), describes totally the singularities of C . It will be proved2hat the factorization of T ( t ) provides the fibre functions of the differentsingularities of C as well as their corresponding multiplicities. The fibrefunction of a point P ∈ C via P ( t ) is given by a polynomial H P ( t ) whichsatisfies that t ∈ F P ( P ) if and only if H P ( t ) = 0. In [13], it is provedthat if H P ( t ) = Q ni =1 ( t − s i ) k i then, C has n tangents at P of multiplicities k , . . . , k n , respectively. In addition, these tangents can be computed using P ( t ) and the roots of each corresponding fibre function. Furthermore, it isshown that mult P ( C ) = deg( H P ( t )) . Taking into account these previous results, in this paper we prove that theT–function can be factorized as T ( t ) = Q ni =1 H P i ( s ) m i − , where H P i ( t ) is thefibre function of the ordinary singularity P i ∈ C and m i is its multiplicity (for i = 1 , . . . , n ). Thus, a complete classification of the singularities of a givenrational curve, via the factorization of a univariate resultant, is obtained.On finishing this work, we just found a paper by Abhyankar (see [1]) thatproves the factorization of the T–function for a given polynomial parametriza-tion. In addition, Bus´e et al., in [5], provide a generalization of Abhyankar’sformula for the case of rational parametrizations (not necessarily polyno-mial). This approach is based on the concept of singular factors introducedin [6], and it involves the construction of µ –basis. Our approach is totallydifferent, since we generalize Abhyankar’s formula by using the methods andtechniques presented in [13]. This allows us to group the factors of the T–function to easily obtain the fibre functions of the different singularities. Inaddition, we show how to deal with singularities that are reached by algebraicvalues of the parameter.As we mentioned above, these results can be stated similarly for the caseof rational space curves in any dimension. We remark that the methods de-veloped in this paper generalize some previous results that partially approachthe computation and analysis of singularities for rational parametrized curves(see e.g. [4], [13] or [17]). Moreover, the ideas presented open several impor-tant ways that may be used to obtain significant results concerning rationalparametrizations of surfaces. In a future work, this problem will be developedin more detail and some important results are expected to be provided.The structure of the paper is as follows. Sections 2 and 3 are devotedto the study of plane curves. In particular, in Section 2, we introduce theterminology that will be used throughout this paper as well as some previousresults. In Section 3, we introduce the T–function and we present the main3esult of the paper. It claims that the factorization of the T–function providesthe fibre functions of the different singularities of the curve. The proof ofthis result as well as some previous technical lemmas appear in Section 5.Section 4 is devoted to generalize the results in Section 3 to parametric spacecurves in any dimension. Throughout the whole paper, we outline all theresults obtained with illustrative examples.
2. Analysis and computation of the fibre
Let C be a rational (projective) plane curve defined by the projectiveparametrization P ( t ) = ( p ( t ) : p ( t ) : p ( t )) ∈ P ( K ( t )) , where gcd( p , p , p ) = 1, and K is an algebraically closed field of characteristiczero . We assume that C is not a line (a line does not have multiple points).Let d = deg( p ), d = deg( p ), d = deg( p ), and d = max { d , d , d } . Thus,we may write p , p and p as p ( t ) = a + a t + a t + · · · + a d t d p ( t ) = b + b t + b t + · · · + b d t d p ( t ) = c + c t + c t + · · · + c d t d . Associated with P ( t ), we consider the induced rational map ψ P : K −→C ⊂ P ( K ); t ( t ) . We denote by deg( ψ P ) the degree of the rational map ψ P (for further details see e.g. [19] pp.143, or [8] pp.80). As an importantresult, we recall that the birationality of ψ P , i.e. the properness of P ( t ),is characterized by deg( ψ P ) = 1 (see [8] and [19]). Also, we recall that thedegree of a rational map can be seen as the cardinality of the fibre of a genericelement (see Theorem 7, pp. 76 in [19]). We will use this characterizationin our reasoning. For this purpose, we denote by F P ( P ) the fibre of a point P ∈ C via the parametrization P ( t ); that is F P ( P ) = P − ( P ) = { t ∈ K | P ( t ) = P } . In general, it holds that P ∈ C if and only if F P ( P ) = ∅ , although anexception can be found for the limit point of the parametrization. Definition 1.
We define the limit point of the parametrization P ( t ) as P L = lim t →∞ P ( t ) /t d = ( a d : b d : c d ) . P L ∈ C since P ( t ) /t d = P ( t ) ∈ C , for t ∈ K , and C is aclosed set. Furthermore, we observe that, given a parametrization P ( t ),there always exists an associated limit point, and it is unique.The limit point is reachable via the parametrization P ( t ), if there exists t ∈ K such that P ( t ) = P L . However, the value t ∈ K could not exist,and then F P ( P L ) = ∅ . Taking into account this statement, if P L is not anaffine point or it is a reachable affine point, we have that P ( t ) is a normalparametrization . Otherwise, we say that P ( t ) is not normal and P L is the critical point (see Subsection 6.3 in [18]). Further properties of the limitpoint are stated and proved in [3].In Subsection 2.2. in [18], it is stated that the degree of a dominantrational map between two varieties of the same dimension is the cardinalityof the fiber of a generic element. Therefore, in the case of the mapping ψ P , this implies that almost all points of C (except at most a finite numberof points) are generated via P ( t ) by the same number of parameter values,and this number is the degree of ψ P . Thus, intuitively speaking, the degreemeasures the number of times the parametrization traces the curve when theparameter takes values in K . Taking into account this intuitive notion, thedegree of the mapping ψ P is also called the tracing index of P ( t ). In orderto compute the tracing index, the following polynomials are considered, G ( s, t ) := p ( s ) p ( t ) − p ( s ) p ( t ) G ( s, t ) := p ( s ) p ( t ) − p ( s ) p ( t ) G ( s, t ) := p ( s ) p ( t ) − p ( s ) p ( t ) (1)and G ( s, t ) = gcd( G ( s, t ) , G ( s, t ) , G ( s, t )). In the following theorem, wecompute the tracing index of P ( t ) using the polynomial G ( s, t ) (see Subsec-tion 4.3 in [18]). Theorem 1.
It holds that deg( ψ P ) = deg t ( G ) . Remark 1.
We observe that: The polynomials G , G and G satisfy that G i ( s, t ) = − G i ( t, s ) . Clearly, G ( s, t ) also has this property. Taking into account the above statement, it holds that deg s ( G i ) =deg t ( G i ) for i = 1 , , , and deg s ( G ) = deg t ( G ) . It holds that deg t ( G ) = max { d , d } . Indeed: if d = d , the statementtrivially holds. If d = d , deg t ( G ) may decrease if p ( s ) c d − p ( s ) a d =5 . But this would imply that C is a line, which is impossible by theassumption. Similarly, it holds that deg t ( G ) = max { d , d } , and deg t ( G ) = max { d , d } . It holds that G ( s, t ) = gcd( G ( s, t ) , G ( s, t )) . Indeed: since p ( t ) G ( s, t ) = p ( t ) G ( s, t ) − p ( t ) G ( s, t ) , if h ( s, t ) ∈ K [ s, t ] divides to G ( s, t ) and G ( s, t ) , then h ( s, t ) divides to G ( s, t ) or p ( t ) . However, if h ( s, t ) divides p ( t ) , then h ( s, t ) = h ( t ) which wouldimply that there exists t ∈ K such that G ( s, t ) = G ( s, t ) = p ( t ) =0 . Hence, p i ( s ) /p ( s ) ∈ K , i = 1 , , and C would be a line, which isimpossible by the assumption. Similarly, it holds that G ( s, t ) = gcd( G ( s, t ) , G ( s, t )) = gcd( G ( s, t ) , G ( s, t )) . Throughout this paper, we assume that P ( t ) is proper, that is deg( ψ P ) =1. Otherwise, we can reparametrize the curve using, for instance, the resultsin [12]. Under these conditions, it holds that the degree of C is d (see Theorem6 in [13]). In addition, G ( t, s ) = t − s (see Theorem 1) and the cardinality ofthe fibre for a generic point of C is 1, although for a particular point it canbe different.In order to analyze these special points, in the following, we consider aparticular point P = ( a, b, c ) ∈ C . The fibre of P consists of the values t ∈ K such that P ( t ) = P , that is, those which satisfy the fibre equations , definedas φ ( t ) := ap ( t ) − cp ( t ) = 0 φ ( t ) := bp ( t ) − cp ( t ) = 0 φ ( t ) := ap ( t ) − bp ( t ) = 0 . (2)Hence, the fibre of P is given by the common roots of these equations,which motivates the following definition: Definition 2.
Given P ∈ P ( K ) and the rational parametrization P ( t ) ∈ P ( K ( t )) , we define the fibre function of P at P ( t ) as H P ( t ) := gcd( φ , φ , φ ) . Thus, t ∈ F P ( P ) if and only if H P ( t ) = 0 . emark 2. Depending on whether P is an affine point or an infinity point,the fibre function can be expressed as follows: • If P is an affine point, then c = 0 . Thus, φ can be obtained from φ and φ and, therefore, H P ( t ) = gcd( φ ( t ) , φ ( t )) . • If P is an infinity point, then c = 0 . Thus, φ and φ are equiva-lent to p ( t ) = 0 (note that a = 0 or b = 0 ) and, therefore, H P ( t ) =gcd( p ( t ) , φ ( t )) . Note that the functions φ , φ and φ depend on P and P ( t ) . However, forthe sake of simplicity, we do not represent this fact in the notation. In the following, we show how the fibre function of P is related with thetangents of C at P , and with the multiplicity of P . For this purpose, we firstrecall that P is a point of multiplicity ℓ on C if and only if all the derivatives of F (where F denotes the implicit polynomial defining C ) up to and includingthose of ( ℓ − P but at least one ℓ − th derivative doesnot vanish at P . We denote it by mult P ( C ). The point P is called a simplepoint on C if and only if mult P ( C ) = 1. If mult P ( C ) = ℓ >
1, then we saythat P is a multiple or singular point (or singularity) of multiplicity ℓ on C or an ℓ –fold point. Clearly P
6∈ C if and only if mult P ( C ) = 0.Observe that the multiplicity of C at P is given as the order of the Taylorexpansion of F at P . The tangents to C at P are the irreducible factorsof the first non–vanishing form in the Taylor expansion of F at P , and themultiplicity of a tangent is the multiplicity of the corresponding factor. Ifall the ℓ tangents at the ℓ -fold point P are different, then this singularity iscalled ordinary , and non–ordinary otherwise. Thus, we say that the characterof P is either ordinary or non-ordinary.In [13], it is shown how to compute the singularities and its correspondingmultiplicities from a given parametrization defining a rational plane curve.Furthermore, it is provided a method for computing the tangents and foranalyzing the non–ordinary singularities. In particular, the following theoremand corollary are proved. Theorem 2.
Let C be a rational algebraic curve defined by a proper para-metrization P ( t ) , with limit point P L . Let P = P L be a point of C and let H P ( t ) = Q ni =1 ( t − s i ) k i be its fibre function (under P ( t ) ). Then, C has n tangents at P of multiplicities k , . . . , k n , respectively. emark 3. It can not be ensured that two different values of t , namely s i and s i , provide different tangents. Thus, we could have a same tangent (at P ( s i ) = P ( s i ) ) of multiplicity k i + k i . Corollary 1.
Let C be a rational algebraic curve defined by a proper para-metrization P ( t ) , with limit point P L . Let P = P L be a point of C and let H P ( t ) be its fibre function (under P ( t ) ). Then, mult P ( C ) = deg( H P ( t )) . Example 1.
Let C be the rational plane curve defined by the projectiveparametrization P ( t ) = ( − t − t − t − t + 7 t + 17 t + 17 t + 6 : t + 1) ∈ P ( C ( t )) . Let us compute H P ( t ) , where P = (0 : 0 : 1) . Since P is an affine point,we can obtain H P ( t ) from φ and φ (see Remark 2). Since φ ( t ) = − t − t − t − and φ ( t ) = 2 t + 14 t + 34 t + 34 t + 12 , we get that H P ( t ) = gcd( φ , φ ) = 2( t + 3)( t + 1) . Figure 1: Triple point with two tangents
Therefore, P ( −
1) = P ( −
3) = P , and applying Theorem 2, we deducethat C has at P two different tangents, one of multiplicity and the otherone of multiplicity . The parametrizations defining these tangents are givenas τ ( t ) = P ( −
3) + P ′ ( − t and τ ( t ) = P ( −
1) + P ′′ ( − t , respectively (see [13]). Note that these tangents are the lines y = x and y = − x (see Figure 1). Finally, we conclude that P is a non–ordinary pointof multiplicity (see Corollary 1). . Resultants and singularities In Section 2, we show that, given a rational proper parametrization, P ( t ),the multiplicity of a given point, P = P L , is the cardinality of the fibre of P ( t ) at P (see Corollary 1). That is, the multiplicity of P = P ( s ) , s ∈ K is given by the cardinality of the set F P ( P ( s )) = { t ∈ K : P ( t ) = P ( s ) } (note that we are assuming that P = P L ). Observe that s ∈ F P ( P ( s ))and hence, the cardinality of F P ( P ( s )) is greater than or equal to 1. Thus, P ( s ) is a singular point if and only if the cardinality of F P ( P ( s )) is greaterthan 1.Taking into account the above statement, in this section, we show howthe different factors of a univariate resultant computed from the polynomials G i ( s, t ) , i = 1 , , , are exactly the fibre functions of the singularities of C .Thus, in particular, the singularities of C and its corresponding multiplicitiesare determined. The idea for the construction of the resultant is that a point P ( s ) ∈ C , s ∈ K , is a singularity if and only if deg( H P ( s ) ( t )) > P ( s ) have more than one common solution).For this purpose, we first assume that P ( s ) is an affine point. Thus,Remark 2 implies that the fibre equations are given by (cid:26) p ( t ) p ( s ) − p ( s ) p ( t ) = 0 p ( t ) p ( s ) − p ( s ) p ( t ) = 0 . Note that this is equivalent to G ( s , t ) = G ( s , t ) = 0, where G ( s, t ) and G ( s, t ) are the polynomials introduced in (1).Then, P ( s ) is a singular point if and only if G ( s , t ) and G ( s , t ) havemore than one common root or, equivalently, if and only if the polynomials G ( s , t ) / ( t − s ) and G ( s , t ) / ( t − s ) have a common root (we note that s is already a root of G ( s , t ) and G ( s , t )). This implies thatRes t (cid:18) G ( s , t ) t − s , G ( s , t ) t − s (cid:19) = 0 . Hence, given the polynomial R ( s ) = Res t (cid:18) G ( s, t ) t − s , G ( s, t ) t − s (cid:19) ,
9f the point P ( s ) is singular, then R ( s ) = 0. In fact, in [1], it is proved thatthis resultant provides the product of the fibre functions of the singularitiesof the curve, in the case that P ( t ) is a polynomial parametrization. A gen-eralization for the case of a given rational parametrization (not necessarilypolynomial) is presented in [5].Thus, R ( s ) can be used to compute the singularities of the curve, butsome problems could appear. First, the values s ∈ K that provide singularpoints are roots of the polynomial R but the reciprocal is not true; i.e. aroot of R may not provide a singular point. In addition, we are assumingthat the singularity is an affine point, but also singularities at infinity haveto be detected.The T–function , that we introduce below, improves the properties of R ( s )and characterizes the singular points of C (affine and at infinity). In order tointroduce it, we need to consider δ i := deg t ( G i ) , λ ij := min { δ i , δ j } , G ∗ i ( s, t ) := G i ( s, t ) t − s ∈ K [ s, t ]and R ij ( s ) := Res t ( G ∗ i , G ∗ j ) ∈ K [ s ] for i, j = 1 , , , i < j. Definition 3.
We define the T–function of the parametrization P ( t ) as T ( s ) = R ( s ) /p ( s ) λ − . In the following we show that this function provides essential informationconcerning the singularities of the given curve C (see Theorem 3). To startwith, the following proposition claims that the T–function can be definedsimilarly from R ( s ) or R ( s ). In addition, in Corollary 2, we prove that T ( s ) is a polynomial. Proposition 1.
It holds that T ( s ) = R ( s ) p ( s ) λ − = R ( s ) p ( s ) λ − = R ( s ) p ( s ) λ − . Proof:
We distinguish two steps to prove the proposition:
Step 1 R ( s ) p ( s ) λ − = R ( s ) p ( s ) λ − . For this purpose, we see the polynomial G ( s, t ) ∈ K [ s, t ] as a polynomial inthe variable t that is, G ( s, t ) ∈ ( K [ s ])[ t ]. Since deg t ( G ) = deg s ( G ) = δ (see Remark 1), then G has δ roots (in the variable t ), and one of them is t = s . Thus, we may write G ( s, t ) = lc t ( G )( t − s )( t − α ( s )) · · · ( t − α δ − ( s ))and G ∗ ( s, t ) = lc t ( G )( t − α ( s )) · · · ( t − α δ − ( s )) , (3)where lc t ( · ) denotes the leader coefficient with respect to the variable t ofa polynomial ( · ). Now, taking into account the properties of the resultants(see e.g. [7], [18], [20]), we get that R ( s ) := Res t ( G ∗ , G ∗ ) = lc t ( G ∗ ) δ − δ − Y i =1 G ∗ ( s, α i ( s )) . (4)Note that G ( s, t ) = p ( s ) p ( t ) − p ( s ) p ( t ), and thus G ( s, α i ( s )) = p ( s ) p ( α i ( s )) − p ( s ) p ( α i ( s )) . Furthermore, since G ( s, α i ( s )) = p ( s ) p ( α i ( s )) − p ( s ) p ( α i ( s )) =0, we get that p ( α i ( s )) = p ( α i ( s )) p ( s ) p ( s ) . Therefore, G ( s, α i ( s )) = p ( s ) p ( α i ( s )) p ( s ) p ( s ) − p ( s ) p ( α i ( s )) =( p ( α i ( s )) p ( s ) − p ( α i ( s )) p ( s )) p ( s ) p ( s ) = G ( s, α i ( s )) p ( s ) p ( s )which implies that G ∗ ( s, α i ( s )) = G ∗ ( s, α i ( s )) p ( s ) p ( s ) . Now we substitute in (4) and we get R ( s ) = lc t ( G ∗ ) δ − δ − Y i =1 G ∗ ( s, α i ( s )) ! (cid:18) p ( s ) p ( s ) (cid:19) δ − , R ( s ) = R ( s )lc t ( G ∗ ) δ − δ (cid:18) p ( s ) p ( s ) (cid:19) δ − . Hence, we only have to prove thatlc t ( G ∗ ) δ − δ (cid:18) p ( s ) p ( s ) (cid:19) δ − = p ( s ) λ − p ( s ) λ − , (5)and, for this purpose, we consider different cases depending on d , d and d (that is, on the degrees of p , p and p ). We remind that δ = max { d , d } , δ = max { d , d } and δ = max { d , d } (see Remark 1). • Case 1: Let d < d . Then, δ = d ≤ δ , and λ = δ . In addition,it holds that lc t ( G ∗ ) = lc t ( G ) = p ( s ) since G ( s, t ) = p ( s ) p ( t ) − p ( s ) p ( t ) and d < d . Thus,lc t ( G ∗ ) δ − δ (cid:18) p ( s ) p ( s ) (cid:19) δ − = p ( s ) δ − p ( s ) δ − δ + δ − = p ( s ) λ − p ( s ) δ − δ + δ − . In order to check that (5) holds, we only have to prove that δ − δ + δ = λ . Let us see that this equality holds in the following situations:a) δ < δ : then, d > d , d which implies that δ = δ = d and λ = δ .b) δ > δ : then, d > d , d which implies that δ = δ = d and λ = δ .c) δ = δ : then, d = max { d , d } which implies that d < d = d and δ = δ = δ . • Case 2: Let d > d . Then, δ = d ≤ δ , which implies that λ = δ .In addition, lc t ( G ∗ ) = p ( s ), and thenlc t ( G ∗ ) δ − δ (cid:18) p ( s ) p ( s ) (cid:19) δ − = p ( s ) δ + δ − δ − p ( s ) δ − = p ( s ) δ + δ − δ − p ( s ) λ − . Thus, we only have to prove that δ + δ − δ = λ . For this purpose,we reason similarly as in Case 1 by considering the following situations: δ < δ , δ > δ and δ = δ . 12 Case 3: Let d = d < d . Then, δ = δ = d , and thus lc t ( G ∗ ) δ − δ =1. In addition, δ ≤ δ = δ , which implies that λ = λ = δ . • Case 4: Let d < d = d . In this case, we have that δ = δ = δ andthen, (5) trivially holds. • Case 5: Let d = d = d . This case is similar to Case 4. Step 2
Let us prove that R ( s ) p ( s ) λ − = R ( s ) p ( s ) λ − . For this purpose, we observe that, up to constants in K \ { } , it holds that R ( s ) = R ( s ). Thus, we may write R ( s ) = R ( s ) = lc t ( G ∗ ) δ − δ − Y i =1 G ∗ ( s, β i ( s ))where G ∗ ( s, t ) = lc t ( G )( t − β ( s )) · · · ( t − β δ − ( s )) . Now, we observe that these equalities are equivalent to (4) and (3), respec-tively. Thus, reasoning similarly as above, we obtain that R ( s ) = R ( s )lc t ( G ∗ ) δ − δ (cid:18) p ( s ) p ( s ) (cid:19) δ − and that lc t ( G ∗ ) δ − δ (cid:18) p ( s ) p ( s ) (cid:19) δ − = p ( s ) λ − p ( s ) λ − . (cid:3) Corollary 2.
It holds that T ( s ) ∈ K [ s ] . Proof:
Let us assume that T ( s ) is not a polynomial. Then, we simplify therational function and we write R ( s ) p ( s ) λ − = M ( s )¯ p ( s ) , M ( s ) ∈ K [ s ], p ( s ) ∈ K [ s ] \ K and gcd( M , ¯ p ) = 1. Note that p divides p λ − .Similarly, from Proposition 1, we have that R ( s ) p ( s ) λ − = M ( s )¯ p ( s ) where p divides p λ − , and gcd( M , ¯ p ) = 1 R ( s ) p ( s ) λ − = M ( s )¯ p ( s ) where p divides p λ − , and gcd( M , ¯ p ) = 1 . Furthermore, we have that (see Proposition 1) M ( s )¯ p ( s ) = M ( s )¯ p ( s ) = M ( s )¯ p ( s )which implies that M ( s )¯ p ( s ) = M ¯ p ( s ) and M ( s )¯ p ( s ) = M ¯ p ( s ) . Taking into account that gcd( M , ¯ p ) = gcd( M , ¯ p ) = gcd( M , ¯ p ) = 1,and the above equalities, we get that p = p = p . Then, we deduce that p divides p , p and p , which is impossible since gcd( p , p , p ) = 1. (cid:3) In the following theorem, we show how the ordinary singularities of C can be determined from the T–function. In fact, T ( s ) describes totally thesingularities of the curve, since its factorization provides the fibre functionsof each singularity as well as its corresponding multiplicity. From the fibrefunction, H P ( t ), of a point P , one obtains the multiplicity of P , its fibre, andthe tangent lines at P (see Section 2).An alternative approach for computing this factorization, based on theconstruction of µ –basis, can be found in [5].In Theorem 3, we assume that C has only ordinary singularities. Other-wise, for applying this theorem, we should apply quadratic transformations(blow-ups) for birationally transforming C into a curve with only ordinarysingularities (see Chapter 2 in [18]). For such a curve the following theoremholds. Theorem 3. (Main theorem)
Let C be a rational algebraic curve definedby a parametrization P ( t ) , with limit point P L . Let P , . . . , P n be the singularpoints of C , with multiplicities m , . . . , m n respectively. Let us assume that hey are ordinary singularities and that P i = P L for i = 1 , . . . , n . Then, itholds that T ( s ) = n Y i =1 H P i ( s ) m i − . This theorem will be proved in Section 5 and a generalization for the caseof space curves of any dimension will be presented in Section 4. Moreover,in [3], we prove that the theorem holds also if P L is a singularity. Finally, ananalogous result which admits the existence of non–ordinary singularities inthe curve will be developed in a future work. Corollary 3.
Let C be a rational plane curve such that all its singularitiesare ordinary. Let P ( t ) be a parametrization of C such that P L is regular. Itholds that deg( T ) = ( d − d − . Proof:
From Theorem 3 and Corollary 1, we have that deg( T ) = P ni =1 m i ( m i − , where P , . . . , P n are the singular points, and m , . . . , m n its correspondingmultiplicities. Since C is a rational curve, its genus is 0 and thus P ni =1 m i ( m i −
1) = ( d − d −
2) (see Chapter 3 in [18]). (cid:3)
Example 2.
Let C be the rational plane curve defined by the projectiveparametrization P ( t ) = ( t − t +5 t +5 t − t : t +5 t +5 t − t − t : t − t +36) ∈ P ( C ( t )) . We compute the T–function and we get that, up to constants in C , T ( s ) = ( t − t − t + 3)( t + 2) t ( t + 6)( t − ( t + 1) . Thus, from Theorem 3, we deduce that the fibre function of each singularityof C appears in the polynomial T . Let us analyze the different factors of T : • The factors with power correspond to triple points. Indeed: thesefactors are t , ( t − and ( t + 1) , and we have that P = P (0) = P (1) = P ( −
1) = (0 : 0 : 1) . Then, P is a triple point whose fibre function is H P ( t ) = ( t − t + 1) t. The factors with power correspond to different double points. In orderto determine the associated factors, we should compute the correspond-ing fibre functions. For the factors ( t − and ( t − , we have that P = P (2) = P (3) = (0 : 1 : 0) and thus H P ( t ) = ( t − t − . This implies that P is a double point at infinity. • Similarly, if we consider the factors ( t + 2) and ( t + 3) , we get that P = P ( −
2) = P ( −
3) = (1 : 0 : 0) , and its fibre function is H P ( t ) = ( t + 2)( t + 3) , which implies that P is a double point at infinity. • Finally, the factor ( t +6) provides the point P = P ( − I √
6) = P (+ I √
6) =( − / / . Hence, P is an affine double point and its fibre func-tion is H P ( t ) = ( t + 6) . Figure 2: Curve C (left) and a neighborhood of the triple point P (right) Note that, T ( s ) = H P ( s ) m − H P ( s ) m − H P ( s ) m − H P ( s ) m − . urthermore, we observe that Corollary 3 is verified. Indeed, we have that d = 5 and deg t ( T ) = 12 = ( d − d − .In Figure 2, we plot the curve C , and a neighborhood of the triple point P . Observe that P is an isolated point. In Example 2, we have been able to determine the singularities of C andits corresponding multiplicities, by computing the factors of the polynomial T ( s ). However, in general, one needs to introduce algebraic numbers duringthe computations. In the following, we present a method that allows usto determine the factors of the polynomial T ( s ) (and thus, the singularitiesof a curve) without directly using algebraic numbers. For this purpose, weintroduce the notion of family of conjugate parametric points (see [13]), whichgeneralizes the concept of family of conjugate points (see e.g. Chapter 2 in[18]). The idea is to collect points whose coordinates depend algebraicallyon all the conjugate roots of the same irreducible polynomial m ( t ). Thecomputations on such a family of points can be carried out by using thepolynomial m ( t ). Definition 4.
Let G = { ( p ( α ) : p ( α ) : p ( α )) | m ( α ) = 0 } ⊂ P ( K ) . The set G is called a family of conjugate parametric points over K if thefollowing conditions are satisfied: p , p , p, m ∈ K [ t ] and gcd( p , p , p ) = 1 . m is irreducible.
3. deg( p ) , deg( p ) , deg( p ) < deg( m ) .We denote such a family by G = {P ( s ) } m ( s ) = { ( p ( s ) : p ( s ) : p ( s )) } m ( s ) . Condition 2 in Definition 4 can be stated considering that m is onlysquare-free (see Definition 12 in [13]). However, in order to prove Theorem4, one needs m to be an irreducible polynomial (see Theorem 16 in [13]).Hence, using the above definition, in [13] it is proved the following theorem. Theorem 4.
The singularities of the curve C can be decomposed as a finiteunion of families of conjugate parametric points over K such that all pointsin the same family have the same multiplicity and character.
17f some singularities of the given curve are in a family G = {P ( s ) } m ( s ) = { ( p ( s ) : p ( s ) : p ( s )) } m ( s ) , then the polynomial m ( s ) will be an irreduciblefactor of the T–function. In this case, Theorem 5 allows us to determine thesingularities provided by G and their corresponding multiplicities. Theorem 5.
Let m ( s ) be an irreducible polynomial such that m ( s ) k − , k ∈ N , k ≥ , divides T ( s ) . Then, the roots of m ( s ) determine the fibre ofsome singularities of multiplicity k that are defined by a family of conju-gate parametric points. The number of singularities in such a family is n = deg( m ( s )) /k . Proof:
From Theorem 3, we get that the points in G = {P ( s ) } m ( s ) aresingularities of multiplicity k . In addition, if G = { P , . . . , P n } , we have that m ( s ) = n Y i =1 H P i ( s ) , where H P , . . . , H P n are the fibre functions of the points P , . . . , P n , respec-tively. From Corollary 1, we get that deg( H P i ) = mult P i ( C ) , i = 1 , . . . , n, and since, in this case, mult P i ( C ) = k , we conclude that deg( m ( s )) = nk. (cid:3) Example 3.
Let C be the rational curve defined by P ( t ) = ( p ( t ) : p ( t ) : p ( t )) ∈ P ( C ( t )) , where p ( t ) = t − t + 63 t − t + 152 t − p ( t ) = t − t + 157 t − t + 706 t − p ( t ) = t + 7 t − t + t − . The T–function is given by T ( s ) = 28161216(968585964 − s + 2070988203 s − s +208513387 s − s + 1145528 s )( s − ( s − ( s − . From the polynomial T ( s ) , we deduce that the singularities of C are: • The triple point P = P (1) = P (2) = P (3) = (0 : 0 : 1) . The fibrefunction of P is H P ( s ) = ( s − s − s − , and these factors appearwith power in the polynomial T ( s ) . Three double points associated to the irreducible factor m ( s ) = 968585964 − s + 2070988203 s − s +208513387 s − s + 1145528 s . Since m ( s ) appears with power in the polynomial T ( s ) , we concludethat it is associated to double points. In addition, using Theorem 5,we get that this factor provides three different points (each point hasa fibre function of degree and the three fibre functions have power ;multiplying these fibre functions we obtain the polynomial m ).In Figure 3, we plot the given curve C , and we can see the singularitiesobtained (note that each singularity is real and affine). Figure 3: The curve C that has a triple point and three double points
4. The general case for rational space curves
In this section, we show that Theorem 3 can be applied for the case thatthe given curve C is a rational space curve. In this case, we construct anequivalent polynomial to the T–function introduced for plane curves (seeDefinition 3), and we prove that this polynomial, which will be denoted as19 E ( s ), describes totally the singularities of C , since each factor of T E ( s ) is apower of the fibre function of one singularity of the given curve. This poweris, in fact, the multiplicity of the singularity minus 1. We recall that fromthe fibre function of a point P , one may determine the multiplicity of P aswell as its fibre F P ( P ) and the tangent lines of C at P (see Section 2). Themethod presented generalizes the results obtained in [17], since a completeclassification of the singularities of a given space curve, via the factorizationof a univariate resultant, is obtained.In the following, we consider P ( t ) = ( p ( t ) : · · · : p n ( t ) : p ( t )) ∈ P n ( K ( t )) , gcd( p , . . . , p n , p ) = 1 , a proper parametrization of a given rational space curve C . In addition,we define the associated rational parametrization over K ( Z ), where Z =( Z , . . . , Z n − ) and Z , . . . , Z n − are new variables, given by b P ( t ) = ( b p ( t ) : b p ( t ) : b p ( t )) == ( p ( t ) : p ( t ) + Z p ( t ) + · · · + Z n − p n ( t ) : p ( t )) ∈ P (( K ( Z ))( t )) . This notation is used for the sake of simplicity, but we note that b P ( t ) dependson Z . Observe that b P ( t ) is a proper parametrization of a rational plane curve b C defined over the algebraic closure of K ( Z ).We can establish a correspondence between the points of C and the pointsof b C . More precisely, for each point P = ( a : a : a : · · · : a n : a n +1 ) ∈ C we have another point b P = ( a : a + Z a + · · · + Z n − a n : a n +1 ) ∈ b C .Moreover, this correspondence is bijective for the points satisfying that a = 0or a n +1 = 0. For these points, it holds that F P ( P ) = F b P ( b P ), which impliesthat H P ( s ) = H b P ( s ). Note that the polynomial H P represents the fibrefunction of a point P in the space curve C computed from P ( t ); i.e. the rootsof H P are the fibre of P ∈ C (this notion was introduced in Definition 2 fora given plane curve but it can be easily generalized for space curves).Observe that the above correspondence may also be established betweenthe places of C and b C centered at b P and P , respectively. That is, for eachplace ϕ ( t ) = ( ϕ ( t ) : ϕ ( t ) : ϕ ( t ) : · · · : ϕ n ( t ) : ϕ n +1 ( t )) of C centered at P wehave the place b ϕ ( t ) = ( ϕ ( t ) : ϕ ( t )+ Z ϕ ( t )+ · · · + Z n − ϕ n ( t ) : ϕ n +1 ( t )) of b C centered at b P . Hence, the number of tangents of C at P is the same that thenumber of tangents of b C at b P and, as a consequence, mult P ( C ) = mult b P ( b C ).20he correspondence above introduced is not bijective if a = a n +1 = 0.In this case, we have that b P = (0 : 1 : 0) ∈ b C and the corresponding pointsin C are all the points of the form (0 : a : a : · · · : a n : 0). Thus, if we haveexactly ℓ points P , . . . , P ℓ ∈ C with P i = (0 : a ,i : a ,i : · · · : a n,i : 0) , i =1 , . . . , ℓ , then F b P ( b P ) = ∪ ℓi =1 F P ( P i ). Hence, H b P ( s ) = Q ℓi =1 H P i ( s ) (note that F P ( P i ) ∩ F P ( P j ) = ∅ if i = j ) and mult b P ( b C ) = P ℓi =1 mult P i ( C ).Thus, in order to study the singularities of C through those of b C , anadditional difficulty arises when C contains two or more points of the form(0 : a : a : · · · : a n : 0). Let us call them bad points . In the following, wemay assume w.l.o.g. that we are not in this case, i.e. C does not have twoor more bad points. Otherwise, we apply a change of coordinates, and weconsider the new parametrization P ∗ ( t ) = ( p ∗ ( t ) : p ( t ) : · · · : p n ( t ) : p ( t ))of the transformed curve C ∗ , where p ∗ = P ni =1 λ i p i , λ i ∈ K . By appro-priately choosing λ , . . . , λ n ∈ K , we have that gcd( p ∗ , p ) = 1 (note thatgcd( p , . . . , p n , p ) = 1) and thus, C ∗ does not have bad points.Finally, we also note that additional points, which can not be written inthe form ( a : a + Z a + · · · + Z n − a n : a n +1 ) , a i ∈ K , i = 1 , . . . , n + 1, mayappear in the curve b C . Such points are obtained as b P ( t ) for t ∈ K ( Z ) \ K and they do not have a correspondence with any point of C .Under these conditions, let b G , b G and b G be the equivalent polynomialsto G , G and G (defined in (1)), constructed from the parametrization b P ( t ).In addition, let b δ i := deg t ( b G i ) and b λ ij := min { b δ i , b δ j } , i, j = 1 , , , i < j , b G ∗ i ( s, t ) := b G i ( s, t ) t − s ∈ ( K [ Z ])[ s, t ] , i = 1 , , , and b R ij ( s ) := Res t ( b G ∗ i , b G ∗ j ) ∈ ( K [ Z ])[ s ] , i, j = 1 , , , i < j. Then, the T–function of the parametrization b P ( t ) is given by b T ( s ) = b R ( s ) / b p ( s ) b λ − . It holds that b T ( s ) ∈ ( K [ Z ])[ s ] (see Corollary 2), and by Proposition 1 we getthat b T ( s ) = b R ( s ) b p ( s ) b λ − = b R ( s ) b p ( s ) b λ − = b R ( s ) b p ( s ) b λ − . b T ( s ) can be used to define anequivalent polynomial to the T–function introduced for plane curves (seeDefinition 3). This polynomial, will be denoted as T E ( s ).Similarly as in the case of plane curves, we assume that the space curve, C ,has only ordinary singularities. The case of space curves with non–ordinarysingularities will be analyzed in a future work and an equivalent theorem toTheorem 6 will be obtained for this case.Finally we remind, that if C has two or more bad points, we considerthe transformed curve C ∗ introduced above. Note that H P ( s ) = H P ∗ ( s ),where P ∈ C is moved to the point P ∗ ∈ C ∗ when the change of coordi-nates is applied. Undoing this change of coordinates, one recovers the initialsingularities P ∈ C . Theorem 6.
Let C be a rational algebraic space curve defined by a parametriza-tion P ( t ) , with limit point P L . Let P , . . . , P n be the singular points of C , withmultiplicities m , . . . , m n respectively. Let us assume that they are ordinarysingularities and that P i = P L , for i = 1 , . . . , n . Then, it holds that T E ( s ) = n Y i =1 H P i ( s ) m i − , where T E ( s ) = Content Z (cid:16) b T ( s ) (cid:17) ∈ K [ s ] , and Content Z ( b T ) represents thecontent of the polynomial b T w.r.t Z . Proof:
From the above statements, we observe that there exists a bijectivecorrespondence between the points b P = ( a : a + Z a + · · · + Z n − a n : a n +1 ) , a i ∈ K , i = 1 , . . . , n + 1 , of b C and the points P = ( a : a : a : · · · : a n : a n +1 ) of C . Consequently, we have that mult b P ( b C ) = mult P ( C ), whichimplies that b P is a singularity of b C of multiplicity m if and only if P is asingularity of C of multiplicity m . Hence, using Theorem 3, we deduce that b T ( s ) = n Y i =1 H P i ( s ) m i − L ( s, Z ) . We observe that the factor L ( s, Z ) ∈ K [ s, Z ] \ K [ s ] is a product of the fibrefunctions corresponding to the singularities of b C that can not be written22s ( a : a + Z a + · · · + Z n − a n : a n +1 ) , a i ∈ K , i = 1 , . . . , n + 1 (thesesingularities do not have an equivalent singularity in C , and its fibre functionnecessarily is a polynomial in K [ s, Z ] \ K [ s ]). Then, we conclude that T E ( s ) = Content Z (cid:16) b T ( s ) (cid:17) = n Y i =1 H P i ( s ) m i − . (cid:3) Example 4.
Let C be the rational space curve defined by the projectiveparametrization P ( t ) = ( p ( t ) : p ( t ) : p ( t ) : p ( t )) ∈ P ( C ( t )) , where p ( t ) = t − t + 5 t + 5 t − tp ( t ) = t + 5 t + 5 t − t − tp ( t ) = t + 7 t + 17 t + 17 t + 6 tp ( t ) = t − t + 36 . We consider the plane curve b P ( t ) = ( p ( t ) : p ( t ) + Zp ( t ) : p ( t )) ∈ P (( C ( Z ))( t )) and compute the corresponding T–function: b T ( s ) = 298598400( s − s − s + 3)( s + 2) s ( s + 1) L ( s, Z ) , where L ( s, Z ) = (3 Z + 1)(2 Z + 1)(25 s + 25 Z s + 50 Zs + 60 Z s + 35 Zs − s +125 s +322 Z s +375 Zs +360 Z s − Zs − s − s − Zs +229 Z s + 150 Zs + 300 Z s + 150 s − Z + 432 Z ) . Removing L ( s, Z ) (whichdepends on Z ), we get T E ( s ) = 298598400( s − s − s + 3)( s + 2) s ( s + 1) . Now, reasoning as in Example 2, we deduce that C has three singularities: • The infinity point P = (0 : 1 : 3 : 0) , with fibre function H P ( t ) =( t − t − (let us remark that P is a bad point; however, it does notrepresent a problem in this case since there are no more bad points in C ). • The infinity point P = (1 : 0 : 0 : 0) , with fibre function H P ( t ) =( t + 2)( t + 3) . • The affine point P = (0 : 0 : 0 : 1) , with fibre function H P ( t ) = t ( t +1) . Note that P , P and P are double points of C (see Figure 4). . . . . − . − .
500 00 − − . . − . − . − . − . − . − . . . − . − . − − . . − . − . − . − . − . − . Figure 4: Curve C and the double point P
5. Proof of the main theorem
This section is devoted to show the main result of this paper, Theorem3 in Section 3. For this purpose, we first prove some previous results. Inparticular, the following lemma is obtained using the main properties of theresultants (see e.g. [7], [18], [20]).
Lemma 1.
Let A ( s, t ) , B ( s, t ) , C ( s, t ) ∈ K [ s, t ] , and K ( s ) ∈ K [ s ] . The fol-lowing properties hold:
1. Res t ( A, K ) = K deg t ( A ) .
2. Res t ( A, B · C ) = Res t ( A, B ) · Res t ( A, C ) . If B divides A , it holds that Res t ( A/B, C ) = Res t ( A, C ) / Res t ( B, C ) .
4. Res t ( A, B + CA ) = lc( A ) k Res t ( A, B ) , where k = deg t ( B + CA ) − deg t B . roof: First, we remind that if A ( t ) , B ( t ) ∈ K [ t ], it holds thatRes t ( A, B ) = lc( A ) deg( B ) Y A ( α i )=0 B ( α i ) . Now, let A ( s, t ) ∈ ( K [ s ])[ t ]. The leader coefficient of A ( s, t ) w.r.t the vari-able t is lc t ( A ) ∈ K [ s ] and, for each s ∈ K , the polynomial A ( s, t ) hasdeg t ( A ) roots, α ( s ) , . . . , α deg t ( A ) ( s ), in the algebraic closure of K ( s ), suchthat A ( s, α i ( s )) = 0, for i = 1 , . . . , deg t ( A ). Then,Res t ( A, B ) = lc t ( A ) deg t ( B ) Y A ( s,α i ( s ))=0 B ( s, α i ( s )) . (6)Now, we prove the four statements of the lemma.1. The first statement follows using (6) for the case that B ( s, t ) ∈ K [ s ].Then deg t ( B ) = 0 and B ( s, α i ( s )) = B ( s ) for each i = 1 , . . . , deg t ( A ).2. In order to prove statement 2, we use (6), and we get thatRes t ( A, B · C ) = lc t ( A ) deg t ( B · C ) Y A ( s,α i ( s ))=0 B ( s, α i ( s )) · C ( s, α i ( s )) . Since deg t ( B · C ) = deg t ( B ) + deg t ( C ), we have thatRes t ( A, B · C ) = lc t ( A ) deg t ( B ) Y A ( s,α i ( s ))=0 B ( s, α i ( s )) lc t ( A ) deg t ( C ) Y A ( s,α i ( s ))=0 C ( s, α i ( s )) = Res t ( A, B ) · Res t ( A, C ) .
3. Reasoning similarly as in statement 2, we get thatRes t ( A/B, C ) = lc t ( A/B ) deg t ( C ) Y A ( s,α i ( s ))=0 , B ( s,α i ( s )) =0 C ( s, α i ( s )) . Since B divides A , we obtain thatlc t ( A/B ) deg t ( C ) Y A ( s,α i ( s ))=0 , B ( s,α i ( s )) =0 C ( s, α i ( s )) =lc t ( A ) deg t ( C ) lc t ( B ) deg t ( C ) Q A ( s,α i ( s ))=0 C ( s, α i ( s )) Q B ( s,α i ( s ))=0 C ( s, α i ( s )) = Res t ( A, C )Res t ( B, C ) .
25. We reason similarly as above, and we get thatRes t ( A, B + CA ) = lc t ( A ) deg t ( B + CA ) · Y A ( s,α i ( s ))=0 ( B ( s, α i ( s )) + C ( s, α i ( s )) A ( s, α i ( s ))) == lc( A ) deg t ( B + CA ) − deg t B lc( A ) deg t ( B ) Y A ( s,α i ( s ))=0 B ( s, α i ( s )) == lc( A ) k Res t ( A, B ) , where k = deg t ( B + CA ) − deg t B. The following lemma provides a first approach to the proof of the mainresult presented in this paper (Theorem 3 in Section 3). In particular, it isshown that each factor H P ( s ) m − , where P is an ordinary singular point ofmultiplicity m , divides the T–function. Lemma 2.
Let C be a rational algebraic curve defined by a parametrization P ( t ) , with limit point P L . Let P = P L be an ordinary singular point ofmultiplicity m . It holds that T ( s ) = H P ( s ) m − T ∗ ( s ) , where T ∗ ( s ) ∈ K [ s ] and gcd( H P ( s ) , T ∗ ( s )) = 1 . Proof:
In order to prove this lemma, we distinguish three different steps. InStep 1, we prove that the lemma holds if P = (0 : 0 : 1). In Step 2, we showthat the lemma holds for any affine singularity. Finally, in Step 3, we provethe lemma for P being a singular point at infinity. Step 1
Let us assume that the given singularity is the point P = (0 : 0 : 1). Notethat H P ( t ) = gcd( φ , φ ) = gcd( p , p ) since, in this case, a = b = 0 (see(2)). Thus, we may write (cid:26) p ( t ) = H P ( t ) p ( t ) p ( t ) = H P ( t ) p ( t ) , where p ( t ) and p ( t ) are polynomials satisfying that gcd( p , p ) = 1. Inaddition, it holds that gcd( H P ( t ) , p ( t )) = 1, since gcd( p , p , p ) = 1. Hence,26rom (1), we may write G ( s, t ) = H P ( s ) H P ( t )( p ( s ) p ( t ) − p ( s ) p ( t )) thatis, G ( s, t ) = H P ( s ) H P ( t ) G ( s, t ) , (7)where G ( s, t ) = p ( s ) p ( t ) − p ( s ) p ( t ).Observe that since P = P L , Corollary 1 holds and then deg( H P ( t )) = m ≥
2. Hence, there exist at least two values s , s ∈ K such that H P ( s ) = H P ( s ) = 0 and then, since these roots belong to the fibre of P , we deducethat (cid:18)(cid:18) p p (cid:19) ( s ) , (cid:18) p p (cid:19) ( s ) (cid:19) = (cid:18)(cid:18) p p (cid:19) ( s ) , (cid:18) p p (cid:19) ( s ) (cid:19) = (0 , . Since P is an ordinary singularity, we have that there can not exist K , K ∈ K such that K (cid:18)(cid:18) p p (cid:19) ′ ( s ) , (cid:18) p p (cid:19) ′ ( s ) (cid:19) = K (cid:18)(cid:18) p p (cid:19) ′ ( s ) , (cid:18) p p (cid:19) ′ ( s ) (cid:19) . (8)We also note that, for i = 1 , j = 0 ,
1, it holds that (cid:18) p i p (cid:19) ′ ( s j ) = p ′ i ( s j ) p ( s j ) − p i ( s j ) p ′ ( s j ) p ( s j ) = p ′ i ( s j ) p ( s j ) , (note that p i ( s j ) = 0). In addition, since p i ( t ) = H P ( t ) p i ( t ), we get that p ′ i ( s j ) p ( s j ) = H ′ P ( s j ) p i ( s j ) + H P ( s j ) p ′ i ( s j ) p ( s j ) = H ′ P ( s j ) p i ( s j ) p ( s j ) , (note that H P ( s j ) = 0). By substituting in (8), we obtain that K (cid:18) H ′ P ( s ) p ( s ) p ( s ) , H ′ P ( s ) p ( s ) p ( s ) (cid:19) = K (cid:18) H ′ P ( s ) p ( s ) p ( s ) , H ′ P ( s ) p ( s ) p ( s ) (cid:19) . Hence, we remark that: • H ′ P ( s i ) = 0 , i = 0 ,
1. That is, s and s are simple roots of H P . • None of the following equalities may be verified: p ( s ) = p ( s ) = 0 p ( s ) = p ( s ) = 0 p ( s ) = p ( s ) = 0 p ( s ) = p ( s ) = 027 If p ( s ) = 0 and p ( s ) = 0, then p ( s ) p ( s ) = p ( s ) p ( s ) . (9)Taking into account these remarks, we prove that T ( s ) = H P ( s ) m − T ∗ ( s ).For this purpose, we first write the T–function as T ( s ) = R ( s ) /p ( s ) λ − (see Proposition 1). From (7), we get that R ( s ) = Res t (cid:16) G ∗ ( s, t ) , H P ( s ) H P ( t ) G ∗ ( s, t ) (cid:17) , where G ∗ ( s, t ) = G ( s, t ) / ( t − s ) (note that G ∗ ( s, t ) ∈ K [ s, t ] since ( t − s )divides G ( s, t )). By applying statement 2 of Lemma 1, we have that R ( s ) =Res t ( G ∗ ( s, t ) , H P ( s )) Res t ( G ∗ ( s, t ) , H P ( t )) Res t (cid:16) G ∗ ( s, t ) , G ∗ ( s, t ) (cid:17) . (10)Let us analyse the first two factors: • From statement 1 of Lemma 1, we have thatRes t ( G ∗ ( s, t ) , H P ( s )) = H P ( s ) deg t ( G ∗ ) = H P ( s ) δ − . • On the other side,Res t ( G ∗ ( s, t ) , H P ( t )) = Res t (cid:18) p ( s ) p ( t ) − p ( s ) p ( t ) t − s , H P ( t ) (cid:19) and, by applying statement 3 of Lemma 1, we get thatRes t ( p ( s ) p ( t ) − p ( s ) p ( t ) , H P ( t ))Res t ( t − s, H P ( t )) . Note that Res t ( t − s, H P ( t )) = H P ( s ) and, since p ( t ) = H P ( t ) p ( t ),the above expression can be written asRes t ( p ( s ) p ( t ) − p ( s ) H P ( t ) p ( t ) , H P ( t )) H P ( s ) . t ( G ∗ ( s, t ) , H P ( t )) = p ( s ) m Res t ( p ( t ) , H P ( t )) lc( H P ( t )) k H P ( s ) , where k ∈ K . Note that, lc( H P ( t )) k ∈ K \ { } and Res t ( p ( t ) , H P ( t )) ∈ K \ { } (since gcd( p, H P ) = 1). Furthermore, we have that p ( s ) = H P ( s ) p ( s ). Thus, up to constants in K \ { } , we deduce thatRes t ( G ∗ ( s, t ) , H P ( t )) = H P ( s ) m − p ( s ) m . Substituting in (10), we obtain that R ( s ) = H P ( s ) δ − H P ( s ) m − p ( s ) m Res t (cid:16) G ∗ ( s, t ) , G ∗ ( s, t ) (cid:17) , and thus T ( s ) = R ( s ) p ( s ) λ − = R ( s ) H P ( s ) λ − p ( s ) λ − = H P ( s ) m − T ∗ ( s ) , where T ∗ ( s ) = Res t (cid:16) G ∗ ( s, t ) , G ∗ ( s, t ) (cid:17) H P ( s ) δ − λ p ( s ) λ − − m . Note that λ = δ since δ ≤ δ . Otherwise, if δ > δ , we would have thatmax { d , d } > max { d , d } and then, d > d , d . However, this would implythat P = P L (see Definition 1), which contradicts the assumption. Therefore, T ∗ ( s ) = Res t (cid:16) G ∗ ( s, t ) , G ∗ ( s, t ) (cid:17) p ( s ) δ − − m . (11)Now, we prove that T ∗ ( s ) ∈ K [ s ]. We can assume that δ − − m ≥
0, since δ = max { d , d } ≥ d and d = deg( p ) = deg( H P · p ) = m + deg( p ).Hence, δ ≥ m + 1 except for the case that deg( p ) = 0, but in this situationwe would have that T ∗ ( s ) = Res t (cid:16) G ∗ ( s, t ) , G ∗ ( s, t ) (cid:17) ∈ K [ s ] . So, let δ − − m ≥
0. Now, we reason similarly as in the proof of Corollary2. Indeed: let us assume that T ∗ ( s ) is not a polynomial. Then, by takingthe simplified rational function, we may write T ∗ ( s ) = M ( s ) b p ( s )29here M ( s ) ∈ K [ s ], b p ( s ) ∈ K [ s ] \ K and gcd( M ( s ) , b p ( s )) = 1. Note that b p ( s ) divides p ( s ) δ − − m .We observe that (11) is obtained from T ( s ) = R ( s ) /p ( s ) λ − . However,taking into account Proposition 1, we could have considered the expression T ( s ) = R ( s ) /p ( s ) λ − concluding that T ( s ) = H P ( s ) m − T ∗ ( s ), where T ∗ ( s ) = Res t (cid:16) G ∗ ( s, t ) , G ∗ ( s, t ) (cid:17) p ( s ) δ − − m . (12)Reasoning similarly as above, we get that there would exist M ( s ) ∈ K [ s ], b p ( s ) ∈ K [ s ] \ K with gcd( M ( s ) , b p ( s )) = 1, such that T ∗ ( s ) = M ( s ) b p ( s ) . In addition, b p ( s ) would divide p ( s ) δ − − m . Thus, we have that M ( s ) b p ( s ) = M ( s ) b p ( s )where gcd( M ( s ) , b p ( s )) = gcd( M ( s ) , b p ( s )) = 1, which implies that b p ( s ) = b p ( s ). Hence gcd( p , p ) = 1, which contradicts the definition of these func-tions.Thus, we have proved that T ∗ is a polynomial. Finally, we show thatgcd( H P ( s ) , T ∗ ( s )) = 1. Indeed: if gcd( H P ( s ) , T ∗ ( s )) = 1, there exists s ∈ K such that H P ( s ) = 0 and T ∗ ( s ) = 0, which implies thatRes t (cid:16) G ∗ ( s, t ) , G ∗ ( s, t ) (cid:17) ( s ) = 0 . Then, by the properties of the resultants, one of the following statementshold:1. There exists s ∈ K such that G ∗ ( s , s ) = G ∗ ( s , s ) = 0. This wouldimply that G ∗ ( s , s ) = 0, and, then, s and s are elements of the fibreof P . On the other side, G ( s , s ) = p ( s ) p ( s ) − p ( s ) p ( s ) = 030nd thus, p ( s ) p ( s ) = p ( s ) p ( s ) . This implies that P is a non–ordinary singular point (see (9)), whichcontradicts the assumptions.2. It holds that gcd(lc t ( G ∗ ) , lc t ( G ∗ ))( s ) = 0. Then, in particular,lc t ( G ∗ )( s ) = lc t ( G )( s ) = p ( s ) c d − p ( s ) a d = 0 ⇒ a d = 0 . Now we reason similarly with the equality T ( s ) = R ( s ) /p ( s ) λ − andwe get (12). From this expression, and reasoning similarly as above,we obtain that gcd(lc t ( G ∗ ) , lc t ( G ∗ ))( s ) = 0, which implies that b d = 0.However, if a d = b d = 0 we deduce that P = P L , which contradicts theassumptions.Therefore, we conclude that gcd( H P ( s ) , T ∗ ( s )) = 1. Step 2
Let P = ( a : b : 1) be a singularity of multiplicity m . In this case, weconsider the translation of the curve C defined by the parametrization e P ( t ) = ( p ( t ) − ap ( t ) : p ( t ) − bp ( t ) : p ( t )) . We have that the point P = ( a : b : 1) moves to the point e P = (0 : 0 : 1),and then H P ( t ) = H e P ( t ) (note that the polynomial H P ( t ) is computed from P ( t ), and the polynomial H e P ( t ) is computed from e P ( t )).On the other side, if we compute the polynomial equivalent to G ( s, t )with the new parametrization e P ( t ), we get that e G ( s, t ) = e p ( s ) e p ( t ) − e p ( s ) e p ( t ) == ( p ( s ) − ap ( s )) p ( t ) − p ( s )( p ( t ) − ap ( t )) = p ( s ) p ( t ) − p ( s ) p ( t ) = G ( s, t ) . Similarly, one obtains that e G ( s, t ) = G ( s, t ). Thus, e R ( s ) = Res t e G ( s, t ) t − s , e G ( s, t ) t − s ! = Res t (cid:18) G ( s, t ) t − s , G ( s, t ) t − s (cid:19) = R ( s ) , e T ( s ) = e R ( s ) / e p ( s ) λ − = R ( s ) /p ( s ) λ − = T ( s ) . Thus, it holds that T ( s ) = H P ( s ) m − T ∗ ( s ) and gcd( H P , T ∗ ) = 1, since fromStep 1, these equalities hold for H e P ( s ) and e T ( s ). Step 3
Let us prove that the lemma holds for a singularity at infinity. For thispurpose, we assume that P = (1 : 0 : 0). Note that we can reason similarlyas in Step 1 taking into account that for this case, H P ( t ) = gcd( p ( t ) , φ ( t )) =gcd( p ( t ) , p ( t )) (see Remark 2). Hence, G ( s, t ) = H P ( s ) H P ( t )( p ( s ) p ( t ) − p ( s ) p ( t )) . and R ( s ) = Res t ( G ∗ , G ∗ ) = Res t (cid:16) G ∗ ( s, t ) , H P ( s ) H P ( t ) G ∗ ( s, t ) (cid:17) where G ∗ ( s, t ) = p ( s ) p ( t ) − p ( s ) p ( t ) t − s . Thus, using the expression T ( s ) = R ( s ) /p ( s ) λ − , we deduce that thelemma also holds if the singularity is the point (1 : 0 : 0). A similar reasoningwith the expression T ( s ) = R ( s ) /p ( s ) λ − shows that the lemma holds forthe point (1 : 0 : 0).Finally, let us assume that P = ( a : b : 0). Then, we reason similarly asin Step 2 and we apply a translation such that the point P is moved to thepoint e P = (1 : 0 : 0). This translation can be defined parametrically by e P ( t ) = ( p ( t )) : p ( t ) − ( b/a ) p ( t ) : p ( t )) . We assume that a = 0; otherwise, it should be b = 0 and we would use atranslation that would move P to (0 : 1 : 0).Under these conditions, we have that H P ( t ) = H e P ( t ). In addition, if wecompute the equivalent polynomials to G ( s, t ) and G ( s, t ) with the newparametrization e P ( t ), we get that e G ( s, t ) = G ( s, t ) and e G ( s, t ) = G ( s, t ).Thus, from Proposition 1, e T ( s ) = e R ( s ) / e p ( s ) λ − = R ( s ) /p ( s ) λ − = T ( s ) . T ( s ) = H P ( s ) m − T ∗ ( s ) and gcd( H P , T ∗ ) = 1, since both equalitieshold for H e P ( s ) and e T ( s ). (cid:3) Proof of the Main Formula (Theorem 3 in Section 3)
Taking into account Lemma 2, we have that for each singular point P i , it holdsthat T ( s ) = H P i ( s ) m i − T ∗ i ( s ), where T ∗ i ( s ) ∈ K [ s ] and gcd( H P i , T ∗ i ) = 1. Inaddition, gcd( H P i , H P j ) = 1 for i = j (otherwise, there would exist s ∈ K such that P ( s ) = P i = P j ). Then, we get that T ( s ) = n Y i =1 H P i ( s ) m i − V ( s ) , where V ( s ) ∈ K [ s ] and gcd( H P i , V ) = 1 for i = 1 , . . . , n .Note that if V ( s ) = 0, then T ( s ) = 0 and thus, R ( s ) = R ( s ) = R ( s ) = 0. From R ( s ) = Res t ( G ∗ ( s, t ) , G ∗ ( s, t ))( s ) = 0 and usingthe properties of the resultant, we deduce that one of the following twostatements hold:1. There exists s ∈ K such that G ∗ ( s , s ) = G ∗ ( s , s ) = 0. Thus, H P ( s ) = 0, where P = P ( s ), which is impossible since gcd( V, H P ) =1.2. It holds that gcd(lc t ( G ∗ ) , lc t ( G ∗ ))( s ) = 0. However, this is also acontraction since we would have thatlc t ( G ∗ )( s ) = lc t ( G )( s ) = p ( s ) c d − p ( s ) a d = 0 ⇒ p ( s ) p ( s ) = a d c d andlc t ( G ∗ )( s ) = lc t ( G )( s ) = p ( s ) b d − p ( s ) a d = 0 ⇒ p ( s ) p ( s ) = a d b d . From both equalities, we deduce that p ( s ) p ( s ) = b d c d , and then P ( s ) = P L . This would imply that P L can be reached bythe parametrization P ( t ) but this implies that it is a singularity (seeProposition 3.4 in [3]), which contradicts the assumption of the theo-rem. 33hus, we have that V ∈ K and, hence, we conclude that, up to constantsin K \ { } , T ( s ) = n Y i =1 H P i ( s ) m i − . (cid:3) References [1] Abhyankar S.S., (1990).
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