Resultants and Singularities of Parametric Curves
Abstract
Let be an algebraic space curve defined parametrically by . In this paper, we introduce a polynomial, the T–function, , which is defined by means of a univariate resultant constructed from . We show that , where are polynomials (called the fibre functions) whose roots are the fibre of the ordinary singularities of multiplicity . Thus, a complete classification of the singularities of a given space curve, via the factorization of a resultant, is obtained.
keywords:
Rational curve parametrization; Singularities of an algebraic curve; Multiplicity of a point; Tangents; Resultant; T–function; Fibre function1 Introduction
Parametrizations of rational curves play an important role in many practical applications in computer aided geometric design where objects are often given and manipulated parametrically (see e.g. Hoffmann (), HSW (), HL97 ()). In the last years, important advances have been made concerning the information one may obtain from a given rational parametrization defining an algebraic variety. For instance, a complete analysis of the asymptotic behavior of a given curve has been carried out in Blas2 (); efficient algorithms for computing the implicit equations that define the curve are provided in Buse2010 () and SWP () and the study and computation of the fibre of a point via the parametrization can be found in SWP (). In addition, some aspects concerning the singularities of the curve and their multiplicities are studied in Abhy (), Buse2012 (), Chen2008 (), JSCPerez () and Rubio (). Similar problems, for the case of a given rational parametric surface, are being analyzed. For instance, the computation of the singularities and their multiplicities from the input parametrization is presented in MACOM (), a univariate resultantbased implicitization algorithm for surfaces is provided in ImpliSuper (), and the computation of the fibre of rational surface parametrizations is developed in FibraSuper ().
In this paper, we show how to relate the fibre and the singularities of a given curve defined parametrically, by means of a univariate resultant which is constructed directly from the parametrization. For this purpose, we consider a rational projective parametrization of an algebraic curve over an algebraically closed field of characteristic zero, . Associated with , we consider the induced rational map We denote by the degree of the rational map . The birationality of , i.e. the properness of , is characterized by (see Harris:algebraic () and shafa ()). Intuitively speaking, proper means that traces the curve once, except for at most a finite number of points. We will see that, in fact, these points are the singularities of .
We recall that the degree of a rational map can be seen as the cardinality of the fibre of a generic element (see shafa ()). We use this characterization in our reasoning and thus, we denote by the fibre of a point via the parametrization ; that is
In order to make the paper more reader–friendly, we first consider the case of a given plane curve defined parametrically by (see Sections 2 and 3) to, afterwards, generalize the results obtained to rational space curves in any dimension (see Section 4). We also assume that has only ordinary singularities (otherwise, one may apply quadratic transformations for birationally transforming the curve into a curve with only ordinary singularities). Non–ordinary singularities have to be treated specially since a non–ordinary singularity might have other singularities in its “neighborhood”. This specific case will be addressed in a future work and in fact, we will show that similar results to those presented in this paper can be stated for curves with non–ordinary singularities.
Under these conditions, the main goal of the paper is to prove that a univariate resultant constructed directly from , which we will call the T–function, , describes totally the singularities of . It will be proved that the factorization of provides the fibre functions of the different singularities of as well as their corresponding multiplicities. The fibre function of a point via is given by a polynomial which satisfies that if and only if . In JSCPerez (), it is proved that if then, has tangents at of multiplicities , respectively. In addition, these tangents can be computed using and the roots of each corresponding fibre function. Furthermore, it is shown that
Taking into account these previous results, in this paper we prove that the T–function can be factorized as , where is the fibre function of the ordinary singularity and is its multiplicity (for ). Thus, a complete classification of the singularities of a given rational curve, via the factorization of a univariate resultant, is obtained.
On finishing this work, we just found a paper by Abhyankar (see Abhy ()) that proves the factorization of the T–function for a given polynomial parametrization. In addition, Busé et al., in Buse2012 (), provide a generalization of Abhyankar’s formula for the case of rational parametrizations (not necessarily polynomial). This approach is based on the concept of singular factors introduced in Chen2008 (), and it involves the construction of –basis. Our approach is totally different, since we generalize Abhyankar’s formula by using the methods and techniques presented in JSCPerez (). This allows us to group the factors of the T–function to easily obtain the fibre functions of the different singularities. In addition, we show how to deal with singularities that are reached by algebraic values of the parameter.
As we mentioned above, these results can be stated similarly for the case of rational space curves in any dimension. We remark that the methods developed in this paper generalize some previous results that partially approach the computation and analysis of singularities for rational parametrized curves (see e.g. Buse2010 (), JSCPerez () or Rubio ()). Moreover, the ideas presented open several important ways that may be used to obtain significant results concerning rational parametrizations of surfaces. In a future work, this problem will be developed in more detail and some important results are expected to be provided.
The structure of the paper is as follows. Sections 2 and 3 are devoted to the study of plane curves. In particular, in Section 2, we introduce the terminology that will be used throughout this paper as well as some previous results. In Section 3, we introduce the T–function and we present the main result of the paper. It claims that the factorization of the T–function provides the fibre functions of the different singularities of the curve. The proof of this 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 space curves in any dimension. Throughout the whole paper, we outline all the results obtained with illustrative examples.
2 Analysis and computation of the fibre
Let be a rational (projective) plane curve defined by the projective parametrization
where , and is an algebraically closed field of characteristic zero . We assume that is not a line (a line does not have multiple points). Let , , , and . Thus, we may write , and as
Associated with , we consider the induced rational map We denote by the degree of the rational map (for further details see e.g. shafa () pp.143, or Harris:algebraic () pp.80). As an important result, we recall that the birationality of , i.e. the properness of , is characterized by (see Harris:algebraic () and shafa ()). Also, we recall that the degree of a rational map can be seen as the cardinality of the fibre of a generic element (see Theorem 7, pp. 76 in shafa ()). We will use this characterization in our reasoning. For this purpose, we denote by the fibre of a point via the parametrization ; that is
In general, it holds that if and only if , although an exception can be found for the limit point of the parametrization.
Definition 1.
We define the limit point of the parametrization as
Note that since , for , and is a closed set. Furthermore, we observe that, given a parametrization , there always exists an associated limit point, and it is unique.
The limit point is reachable via the parametrization , if there exists such that . However, the value could not exist, and then . Taking into account this statement, if is not an affine point or it is a reachable affine point, we have that is a normal parametrization. Otherwise, we say that is not normal and is the critical point (see Subsection 6.3 in SWP ()). Further properties of the limit point are stated and proved in MyB2017(b) ().
In Subsection 2.2. in SWP (), it is stated that the degree of a dominant rational map between two varieties of the same dimension is the cardinality of the fiber of a generic element. Therefore, in the case of the mapping , this implies that almost all points of (except at most a finite number of points) are generated via by the same number of parameter values, and this number is the degree of . Thus, intuitively speaking, the degree measures the number of times the parametrization traces the curve when the parameter takes values in . Taking into account this intuitive notion, the degree of the mapping is also called the tracing index of . In order to compute the tracing index, the following polynomials are considered,
(1) 
and . In the following theorem, we compute the tracing index of using the polynomial (see Subsection 4.3 in SWP ()).
Theorem 1.
It holds that
Remark 1.
We observe that:

The polynomials , and satisfy that . Clearly, also has this property.

Taking into account the above statement, it holds that for , and .

It holds that . Indeed: if , the statement trivially holds. If , may decrease if . But this would imply that is a line, which is impossible by the assumption. Similarly, it holds that , and .

It holds that
Indeed: since , if divides to and , then divides to or . However, if divides , then which would imply that there exists such that . Hence, and would be a line, which is impossible by the assumption. Similarly, it holds that
Throughout this paper, we assume that is proper, that is . Otherwise, we can reparametrize the curve using, for instance, the results in PerezRepara1 (). Under these conditions, it holds that the degree of is (see Theorem 6 in JSCPerez ()). In addition, (see Theorem 1) and the cardinality of the fibre for a generic point of is , although for a particular point it can be different.
In order to analyze these special points, in the following, we consider a particular point . The fibre of consists of the values such that , that is, those which satisfy the fibre equations, defined as
(2) 
Hence, the fibre of is given by the common roots of these equations, which motivates the following definition:
Definition 2.
Given and the rational parametrization , we define the fibre function of at as
Thus, if and only if .
Remark 2.
Depending on whether is an affine point or an infinity point, the fibre function can be expressed as follows:

If is an affine point, then . Thus, can be obtained from and and, therefore,

If is an infinity point, then . Thus, and are equivalent to (note that or ) and, therefore,
Note that the functions , and depend on and . However, for the sake of simplicity, we do not represent this fact in the notation.
In the following, we show how the fibre function of is related with the tangents of at , and with the multiplicity of . For this purpose, we first recall that is a point of multiplicity on if and only if all the derivatives of (where denotes the implicit polynomial defining ) up to and including those of –th order, vanish at but at least one th derivative does not vanish at . We denote it by . The point is called a simple point on if and only if . If , then we say that is a multiple or singular point (or singularity) of multiplicity on or an –fold point. Clearly if and only if .
Observe that the multiplicity of at is given as the order of the Taylor expansion of at . The tangents to at are the irreducible factors of the first non–vanishing form in the Taylor expansion of at , and the multiplicity of a tangent is the multiplicity of the corresponding factor. If all the tangents at the fold point are different, then this singularity is called ordinary, and non–ordinary otherwise. Thus, we say that the character of is either ordinary or nonordinary.
In JSCPerez (), it is shown how to compute the singularities and its corresponding multiplicities from a given parametrization defining a rational plane curve. Furthermore, it is provided a method for computing the tangents and for analyzing the non–ordinary singularities. In particular, the following theorem and corollary are proved.
Theorem 2.
Let be a rational algebraic curve defined by a proper parametrization , with limit point . Let be a point of and let be its fibre function (under ). Then, has tangents at of multiplicities , respectively.
Remark 3.
It can not be ensured that two different values of , namely and , provide different tangents. Thus, we could have a same tangent (at ) of multiplicity .
Corollary 1.
Let be a rational algebraic curve defined by a proper parametrization , with limit point . Let be a point of and let be its fibre function (under ). Then,
Example 1.
Let be the rational plane curve defined by the projective parametrization
Let us compute , where . Since is an affine point, we can obtain from and (see Remark 2). Since and , we get that
Therefore, , and applying Theorem 2, we deduce that has at two different tangents, one of multiplicity and the other one of multiplicity . The parametrizations defining these tangents are given as
respectively (see JSCPerez ()). Note that these tangents are the lines and (see Figure 1). Finally, we conclude that is a non–ordinary point of multiplicity (see Corollary 1).
3 Resultants and singularities
In Section 2, we show that, given a rational proper parametrization, , the multiplicity of a given point, , is the cardinality of the fibre of at (see Corollary 1). That is, the multiplicity of is given by the cardinality of the set
(note that we are assuming that ). Observe that and hence, the cardinality of is greater than or equal to . Thus, is a singular point if and only if the cardinality of is greater than .
Taking into account the above statement, in this section, we show how the different factors of a univariate resultant computed from the polynomials are exactly the fibre functions of the singularities of . Thus, in particular, the singularities of and its corresponding multiplicities are determined. The idea for the construction of the resultant is that a point is a singularity if and only if (i.e. the fibre equations of have more than one common solution).
For this purpose, we first assume that is an affine point. Thus, Remark 2 implies that the fibre equations are given by
Note that this is equivalent to , where and are the polynomials introduced in (1).
Then, is a singular point if and only if and have more than one common root or, equivalently, if and only if the polynomials and have a common root (we note that is already a root of and ). This implies that
Hence, given the polynomial
if the point is singular, then . In fact, in Abhy (), it is proved that this resultant provides the product of the fibre functions of the singularities of the curve, in the case that is a polynomial parametrization. A generalization for the case of a given rational parametrization (not necessarily polynomial) is presented in Buse2012 ().
Thus, can be used to compute the singularities of the curve, but some problems could appear. First, the values that provide singular points are roots of the polynomial but the reciprocal is not true; i.e. a root of may not provide a singular point. In addition, we are assuming that the singularity is an affine point, but also singularities at infinity have to be detected.
The T–function, that we introduce below, improves the properties of and characterizes the singular points of (affine and at infinity). In order to introduce it, we need to consider
and
Definition 3.
We define the T–function of the parametrization as
In the following we show that this function provides essential information concerning the singularities of the given curve (see Theorem 3). To start with, the following proposition claims that the T–function can be defined similarly from or . In addition, in Corollary 2, we prove that is a polynomial.
Proposition 1.
It holds that
Proof: We distinguish two steps to prove the proposition:
Step 1
First, we show that
For this purpose, we see the polynomial as a polynomial in the variable that is, . Since (see Remark 1), then has roots (in the variable ), and one of them is . Thus, we may write
and
(3) 
where denotes the leader coefficient with respect to the variable of a polynomial . Now, taking into account the properties of the resultants (see e.g. Cox1998 (), SWP (), Vander ()), we get that
(4) 
Note that , and thus Furthermore, since , we get that
Therefore,
which implies that
Now we substitute in (4) and we get
which can be expressed as
Hence, we only have to prove that
(5) 
and, for this purpose, we consider different cases depending on , and (that is, on the degrees of , and ). We remind that , and (see Remark 1).

Case 1: Let . Then, , and . In addition, it holds that since and . Thus,
In order to check that (5) holds, we only have to prove that . Let us see that this equality holds in the following situations:

: then, which implies that and .

: then, which implies that and .

: then, which implies that and .


Case 2: Let . Then, , which implies that . In addition, , and then
Thus, we only have to prove that . For this purpose, we reason similarly as in Case 1 by considering the following situations: , and .

Case 3: Let . Then, , and thus . In addition, , which implies that .

Case 4: Let . In this case, we have that and then, (5) trivially holds.

Case 5: Let . This case is similar to Case 4.
Step 2
Let us prove that
For this purpose, we observe that, up to constants in , it holds that . Thus, we may write
where
Now, we observe that these equalities are equivalent to (4) and (3), respectively. Thus, reasoning similarly as above, we obtain that
and that
Corollary 2.
It holds that .
Proof: Let us assume that is not a polynomial. Then, we simplify the rational function and we write
where , and . Note that divides .
Similarly, from Proposition 1, we have that
Furthermore, we have that (see Proposition 1)
which implies that
Taking into account that , and the above equalities, we get that . Then, we deduce that divides , and , which is impossible since .
In the following theorem, we show how the ordinary singularities of can be determined from the T–function. In fact, describes totally the singularities of the curve, since its factorization provides the fibre functions of each singularity as well as its corresponding multiplicity. From the fibre function, , of a point , one obtains the multiplicity of , its fibre, and the tangent lines at (see Section 2).
An alternative approach for computing this factorization, based on the construction of –basis, can be found in Buse2012 ().
In Theorem 3, we assume that has only ordinary singularities. Otherwise, for applying this theorem, we should apply quadratic transformations (blowups) for birationally transforming into a curve with only ordinary singularities (see Chapter 2 in SWP ()). For such a curve the following theorem holds.
Theorem 3.
(Main theorem) Let be a rational algebraic curve defined by a parametrization , with limit point . Let be the singular points of , with multiplicities respectively. Let us assume that they are ordinary singularities and that for . Then, it holds that
This theorem will be proved in Section 5 and a generalization for the case of space curves of any dimension will be presented in Section 4. Moreover, in MyB2017(b) (), we prove that the theorem holds also if is a singularity. Finally, an analogous result which admits the existence of non–ordinary singularities in the curve will be developed in a future work.
Corollary 3.
Let be a rational plane curve such that all its singularities are ordinary. Let be a parametrization of such that is regular. It holds that