Abstract

In analytic geometry, Bézout’s theorem stated the number of intersection points of two algebraic curves and Fulton introduced the intersection multiplicity of two curves at some point in local case. It is meaningful to give the exact expression of the intersection multiplicity of two curves at some point. In this paper, we mainly express the intersection multiplicity of two curves at some point in and under fold point, where . First, we give a sufficient and necessary condition for the coincidence of the intersection multiplicity of two curves at some point and the smallest degree of the terms of these two curves in . Furthermore, we show that two different definitions of intersection multiplicity of two curves at a point in are equivalent and then give the exact expression of the intersection multiplicity of two curves at some point in under fold point.

1. Introduction

Analytic geometry or Cartesian geometry is an important branch of algebra, a great invention of Fermat and Descartes, which deals with the modelling of some geometrical objects, such as points, lines, and curves. It is a mathematical subject that uses algebraic symbolism and methods to solve some geometric problems. It establishes the correspondence between the algebraic equations and the geometric curves. In analytic geometry, one of the most important fundamental problems is to find the number of intersection points of two algebraic curves. Bézout’s theorem stated that two algebraic curves of degrees and intersect in points counting multiplicities and cannot meet in more than points unless they have a component in common ([1]). In a local case, Fulton ([2], Section 3.3) introduced the intersection multiplicity of two affine algebraic curves at some point. Consequently, many mathematicians focussed on the geometric modelling with curves (cf. [3–12]). In this paper, we consider the intersection multiplicities of two curves at some point in and with , respectively, and we mainly give the exact expression of the intersection multiplicities of two curves at some point in and .

Let and be two algebraic curves in , and the intersection multiplicity of and at a point is the number of times that the curves and intersect at the point ([4], Chapter 1). Also, there are some other definitions of the intersection multiplicity of algebraic curves at a point (cf. [2, 7, 11, 13–15]). According to the definitions of and projective transformation, we can transfer some difficult cases to some easy cases by projective transformation when we consider the intersection multiplicity in ; thus, we can connect the intersection multiplicity in with the intersection multiplicity in by homogenizing polynomials (([4], Theorem 3.7), ([16], Lemma 2.5)). Following the factorization theorem of polynomials, we note that the intersection multiplicity of two curves at a point has a close relation with the fold point, so it is important to give the relation between and the fold point. However, for the case of the affine space , where is an algebraically closed field with , we have two different definitions of intersection multiplicity of two curves at a point: one is given by using independent polynomials (([4], Definition 13.2)), and the other is given by means of the dimension of a local ring (([13], Definition 2.3), ([2], Section 3.3)). So it is meaningful to show that the two different definitions are equivalent. Furthermore, similar as the projective transformation, we use affine transformation to extend the intersection multiplicity of curves at some point from to .

This paper is organized as follows: In Section 2 we first introduce some properties of the intersection multiplicity of algebraic curves at some point in and , respectively, and then give a sufficient and necessary condition for the coincidence of the intersection multiplicities of two curves under fold point and the smallest degree of the terms of these two curves in . In generalization, we consider the intersection multiplicity of affine algebraic curves at some point in in Section 3 and we show an equivalence for two different definitions of intersection multiplicity of algebraic curves in . Furthermore, we give some properties for the intersection multiplicity between curves and lines in by using localization and then give some related results about the intersection multiplicity of curves in terms of fold point by using the affine transformation.

2. Intersection Multiplicity of Algebraic Curves in and

In this section, we introduce some properties of the intersection multiplicity of algebraic curves at some point in and the real projective plane , respectively, where is and is the equivalence relation defined by if there exists a nonzero , such that . Consequently, according to the properties of the intersection multiplicities of curves under fold point in , we give a sufficient and necessary condition for the coincidence of the intersection multiplicities of two curves under fold point and the smallest degree of the terms of these two curves in .

2.1. Properties of Intersection Multiplicity of Curves in and

An algebraic curve in is the graph of a polynomial in over . Let and be algebraic curves (abbreviate to curves) which intersect at a point in . The intersection multiplicity of and at is the number of times that the curves and intersect at the point , denoted by .

Property 1 ((see [4], Section 1)). Let , , and be curves and a point in . Then(1) is a nonnegative integer or , and .(2) if and only if and .(3).(4), and if .An algebraic curve (simply curve) in is a homogeneous polynomial in . We can extend algebraic curves from to by homogenizing polynomials. Note that, for any homogeneous polynomial , we set ; then, any point lies on the curve if and only if the point lies on the curve . Let be curves in . Similar as the the definition of , we denote the intersection multiplicity of and at the point . It is clear that have the similar properties as in Property 1.

Lemma 1 ((see [4], Theorem 3.7)). Let and be curves in , and we set . Then, for any point , we have

A projective transformation is a linear map defined bywhere is an invertible matrix. It follows from ([4], Theorem 3.4) that a projective transformation can transform any four points, no three of which are collinear, into any other four such points. Furthermore, projective transformations preserve intersection multiplicities (Lemma 2), and it follows that we can find the intersection multiplicity for any point by transforming to the origin .

Lemma 2 ((see [4], Property 3.5)). Let and be curves and a point in , and is a projective transformation that maps (resp.,) to (resp.,). Then,

2.2. Intersection Multiplicity of Curves under Fold Point in and

Let be a nonzero polynomial and be the sum of the terms of degree in . Then, we can writefor distinct lines uniquely, where is a nonnegative integer and is a polynomial that has no line factors. According to the factorization of , to study the intersection multiplicity of curves in , it is important and convenient to study the intersection multiplicity between curves and lines first.

Lemma 3 ((see [16], Corollary 2.3)). Let be a curve that contains the origin , and is the smallest degree of the terms in . We assume that is a line through the origin . Then, if is a factor of , and if is not a factor of .

Definition 1 ((see [4], §4)). Let be a curve and a point in . We say that the point is a -fold point of if there is a nonnegative integer , such that there are at most distinct lines that intersect at more than times and that all other lines intersect at exactly times.
It follows from Lemma 3 and Definition that the origin is a -fold point of if and only if is the smallest degree of the terms in . Similarly, we can define the fold point of a curve in and have the following Lemma.

Lemma 4 ((see [16], Theorem 3.4)). Let and be curves and a point in . Assume that is a -fold point of and an -fold point of , then

By Lemma 1 and the definition of the projective plane , we can consider the intersection multiplicity of curves under fold point in by setting .

Corollary 1 ((see [16], Corollary 3.5)). Let and be curves and a point in . We assume that is a -fold point of and an -fold point of . Then,

Theorem 1. Let and be curves that both contain the origin in . We assume that the origin is a 1-fold point of both and . Let (.,) be the sum of the terms of degree 1 in (.,). Then, if and only if and have a common factor. Equivalently, if and only if and have no common factors.

Proof. Since the origin is a 1-fold point of , by Lemma 3, we can writefor some polynomials and in . Note that and are not all zeros, and we may assume that . Similarly, we can writefor some polynomials and in with . By Property 1, we haveIt follows that if and only if contains the origin , i.e., the determinant of the matrix is zero. This is equivalent to the condition that and have a common factor. Hence, we obtain the assertion.

Theorem 2. Let and be curves in . We assume that the origin is a -fold point of and an -fold point of . Let (., ) be the sum of the terms of degree (., ) in (., ). Then, if and only if and have a common factor of positive degrees. Equivalently, if and only if and have no common factors of positive degrees.

Proof. Let denote the assertion that if the origin is a -fold point of and an -fold point of and that and have a common factor of positive degrees. When and , it is obvious that holds following Theorem 1.
First, assume that holds for . We prove that also holds. We assume that the origin is a -fold point of and that and have a common factor of positive degrees. By Lemma 3, we can writefor some polynomials and . Note that and are not all zeros. Also, we can writefor some polynomials and , and every term of has degree at least . By linear transformation, we may assume that and . From Property 1, we haveWe set . Then the origin is an -fold point of for following Lemma 3. Let be the sum of the terms of degree in . If , thenwhich implies that and have a common factor of positive degree, and by assumption, holds. If , then from Corollary 1, and we have that holds clearly. By induction, we obtain that holds for any .
Second, we assume that holds for some positive integers and satisfying and . We prove that also holds. We assume that the origin is a -fold point of and that and have a common factor of positive degree. Then, we can writefor some polynomials and , and every term of (resp., ) has degree at least (resp., ). Note that and (resp., and ) are not all zeros. By linear transformation, we may assume that and .
We assume that (similarly for the case ), and following Property 1, we haveWe set . Then, the origin is an -fold point of for . Let be the sum of the terms of degree in . If , thenwhich implies that and have a common factor of positive degrees, and by assumption, we obtain that holds. If , thenfollowing Corollary 1, and clearly we have that holds. Thus, we obtain that holds. Therefore, by induction, we proved that the sufficient condition holds.
Conversely, following Corollary 1, it is suffice to show that if and have no common factors of positive degree, then .
Let denote the assertion that if the origin is a -fold point of and an -fold point of and that and have no common factors of positive degrees. When and , it is clear that holds following Theorem 1.
First, we assume that holds for . We prove that also holds. We suppose that the origin is a -fold point of and that and have no common factors of positive degree. By Lemma 3, we can writefor some polynomials and . Note that and are not all zeros. Also, we can writefor some polynomials and , and every term of has a degree at least . By linear transformation, we may assume that and . Following Property 1, we haveWe set . Thenthe origin is a -fold point of for from Lemma 3. Let be the sum of the terms of degree in . If , thenwhich implies that and have no common factors of positive degree, and by the assumption, we know that holds. If , thenwhich implies that and have a common factor of positive degree. This is a contradiction. Therefore, by induction, we prove that holds for any .
Second, we assume that holds for some positive integers satisfying and . We prove that also holds. We suppose that the origin is a -fold point of and that and have no common factors of positive degrees. Following Lemma 3, we can writefor some polynomials , , , and , and every term of (resp., ) has a degree at least (resp., ). Note that and (resp., and ) are not all zeros. By linear transformation, we may assume that and .
If (similarly for the case ), following Property 1, we haveWe set . Then, the origin is a -fold point of for by Lemma 3. Let be the sum of the terms of degree in . If , thenwhich implies that and have no common factor of positive degree, and holds by the assumption. If , thenwhich implies that and have a common factor of positive degree. This is a contradiction.
Therefore, by induction, we proved that the necessary condition holds.

Example 1. We compute the intersection multiplicity of two algebraic curves and in , where and have a common factor . Note that the origin is a 1-fold point of and a 2-fold point of . According to Theorem 2, we have that . In fact, by Property 1, we have

Example 2. We evaluate the intersection multiplicity of two algebraic curves and in , where and have no common factors. Note that the origin is a 2-fold point of and a 2-fold point of . According to Theorem 2, we have that . In fact, following Property 1, we also have

3. Intersection Multiplicity of Affine Curves in

Following the structure of the affine plane with , it is important to consider the intersection multiplicity of affine curves (abbreviate to curves) at some point in . Since there are two different definitions of intersection multiplicity of curves at some point in , in this section, we first prove that the two different definitions of intersection multiplicity are equivalent. Also, in general, it is not easy to give the intersection multiplicity of curves in , and we use the affine transformation to transfer the difficult case to easy case when we consider the intersection multiplicity of curves under fold point in . In this section, we mainly extend the intersection multiplicity of curves at some point from to . For the case with the prime number , the intersection multiplicity of curves at some point in is still an open question.

3.1. Equivalent Definitions of Intersection Multiplicity of Curves in

Let be an algebraically closed field with , and is the affine -space over . In , we have two different definitions (([13], Definition 2.3), ([4], Definition 13.2), ([2], Section 3.3)) of intersection multiplicity of curves at a point ; however, the intersection multiplicities of curves have the same properties (see ([4], Section 13), ([2], Section 3.3)), so it is important to show that the two different definitions of intersection multiplicity are equivalent.

Definition 2 ((see [4], Definition 13.1)). Let be a positive integer, and let , , and , , be polynomials in and . We call that , , are dependent with respect to and at the point if there are polynomials , , , and , , such thatwhere and , , are not all zeros. In other words, are independent with respect to and at the point if they do not satisfy any equation of the form (29), where and , , are not all zeros.
Using independent polynomials, we can determine the intersection multiplicities of affine algebraic curves in at any point .

Definition 3 ((see [4], Definition 13.2)). Let be polynomials in and . The intersection multiplicity of and at , denoted by , is defined as follows: if is the largest integer such that there are polynomials which are independent with respect to and at ; if for every positive integer , there are at least polynomials which are independent with respect to and at .
Let be the local ring of at a point which is defined asNote that is a vector space under for some polynomials and that is an ideal of .

Definition 4. ((see [13], Definition 2.3), ([2], Theorem 3)). Let , , and . The intersection multiplicity of and at is the dimension of the vector space , denoted byFrom Definitions 3 and 4, we have the following theorem which shows that the two different definitions of intersection multiplicity are equivalent.

Theorem 3. Let , , and , and then, .

Proof. We assume that , and following Definition 3, we know that is the largest number of polynomials which are independent with respect to and at . We suppose that , , are polynomials which are independent with respect to and at . Thenfor any other polynomial ,are dependent with respect to and at .
Let be a map defined byFor any , , if , thenwhich follows thatThis means thatTherefore, is well defined. Thus, by Definition 4, it is suffice to prove that is a basis of the vector space over .(1)First, we prove that are linearly independent over . We suppose that there exist such that that is, It follows that Thus, there exist with such that that is, Since are independent with respect to and at and , we have by Definition 2, which tells that are linearly independent over .(2)Second, we prove that are the generators of the vector space ; that is, any is a linear combination of for any with .Since for any polynomial with ,are dependent with respect to and at , so there are polynomialsand , , such thatwhere and by Definition 2. Thus, we haveSince , we haveIn fact, if , then following the fact that are dependent with respect to and at . This is a contradiction.
Therefore, we obtain that is a basis of the vector space over , that is,In general, it is difficult to give the intersection multiplicity of two curves at some point in . Since the intersection multiplicity is a geometry problem, according to the fact that the properties of figures of geometry are invariant under affine transformations which preserve the intersection multiplicity, we should try to transfer the intersection multiplicity to another easy case by using an affine transformation.
An affine transformation of is a map defined bywhere is a invertible matrix, i.e., , and . Note that an affine transformation preserves collinearity, i.e., all points lying on a line initially still lie on a line after the affine transformation, and the images of two curves intersecting at a point intersect at the point under an affine transformation . In fact, every affine transformation can be represented as the composition of a linear transformation and a translation.
Following ([2], Section 3.3), we have the properties of the intersection multiplicity of curves at the point , which are same as the properties of the intersection multiplicity .

Property 2. Let and be curves in and . Then, we have the following:(1) is a nonnegative integer or .(2) if and only if and both contain the point .(3)Let be an affine transformation that maps (resp., ) to (resp., ). Then,(4).(5).(6).

3.2. Intersection Multiplicity between Curves and Lines in

To consider the intersection multiplicities of curves in with , it is important to consider some properties of intersection multiplicity between curves and lines in . Let be an affine algebraic curve in , and we can write thatwhere is the sum of the terms of degree in . Let be the smallest degree of the terms of with respect to the origin (0,0), and if , then can be written asfor distinct lines , where is a positive integer.

According to Definition 4, in order to complete the proof of Lemma 5, we should introduce the localization which is a very powerful technique in commutative algebra, and localization always allows us to reduce questions on rings and modules to a union of smaller local problems.

Definition 5 ((see [17], Lecture 9)). Let be an integral domain, and is a prime ideal in . The localization of at , denoted by , is defined aswhere is the equivalent class under the equivalence relation which is defined by if there exists such that .
Note that the equivalence relation owned the addition operation:and the multiplication operation isand the localization is a local ring with a maximal ideal . For the integral domain , we have the localization of at the maximal ideal ; i.e., , and is a local ring with the unique maximal ideal .

Lemma 5. Let be a curve, and is the origin in . We assume that and that is a line through the origin . Then, if is a factor of ; otherwise, .

Proof. From Definition 4, we have

Case 1. We suppose that is not a factor of . Since is a prime ideal in , from Definitions 4 and 5, we havewherewhich have addition operationand multiplication operation isSimilarly, we haveNote that and is an ideal of . If , we can write thatThus, we have and . It tells thatSince is a line through the origin in , we can write aswhere and are not all zeros. We may assume that is nonzero, then the line is . Thus, we haveIn fact, letbe the map defined byLet . For anyif , that is,which implies that there are such thatand then, under the map , we havewhich follows thatHence, we havewhich means that the map is well defined.
Furthermore, for any , it is clear thatwhich tells that the map is a ring homomorphism.
On the other hand, if , that is,which implies thatand thus, for any , we haveConsequently, under the map , we havewhich means that . This tells that the map is injective. It is obvious that the map is surjective. Therefore, we haveFollowing (62), we haveNext, we prove that . Since and are the quotient ring of , we know that the following diagram is commutative.

Naturally, we can letbe the map defined byIt is clear that is well defined. To prove , it is suffice to show that is a basis of over .(i)First, are linearly independent over . In fact, we assume that there exist such that that is, It follows that This means that there exists such that which tells that Since are linearly independent over , following Definition 2 we have that are independent with respect to at 0 over .(ii)Second, we have that generate the vector space . In other words, any is a linear combination of , where .By Definition 2, we know that for any polynomial ,are dependent with respect to at 0, so there are polynomials , , and such thatwhere and . Thus, we haveSince , we haveIn fact, if , according to the fact that are independent with respect to at 0 over , we have , which is a contradiction.
Therefore, is a basis of the vector space over . Hence, we haveFollowing (56) and (79), we have

Case 2. If is a factor of , by Property 2, we haveSince , according to the above proof, we have easily. Therefore, we obtain the assertion.

3.3. Intersection Multiplicity of Curves under Fold Point in

Similar as in Section 2.2, we still have some similar results about the intersection multiplicities of curves under fold point in . For the fold point, Hirschfeld et al. (([18], Definition 1.8)) also give the definition of the -fold point of an affine curve. In this paper, we still use the similar definition of fold point as in Definition 1.

Definition 6. Let be a curve and a point in . We say that the point is a -fold point of if there is a nonnegative integer , such that there are at most distinct lines that intersect at more than times and that all other lines intersect at exactly times.
By Property2 we know that the affine transformations preserve the intersection multiplicity of curves in . Following Definition 6, we note that the definition of the fold point depends on lines, so it is a natural question to ask whether the intersection multiplicity between curves and lines in can be transformed into a simple case. We give the following lemma which is helpful in completing the proof of Theorem 4.

Lemma 6. Let be a curve and a point in . If is a -fold point of , then there exist a line through the point and an affine transformation , such thatwhere ,, and .

Proof. Since is a -fold point of and is finite, by Lemma 5, we know that there exist infinite lines through the point such that . It is obvious that there exist an affine transformation and some with , such that and . According to Property 2, it states that the affine transformation preserves the intersection multiplicity of curves in . Thus, we obtain the assertion.

Example 3. Let be an algebraic curve, and we take . Thus, we have , and we can writeSince is the only line such that , any other line which is different from the line satisfies . Thus, by Definition 6, we know that is a 1-fold point of .
Taking , we have from Lemma 5. Since affine transformations preserve intersection multiplicity of curves, our aim is to find an affine transformation such thatBy the proof of Lemma 6, we can easy give the affine transformation defined bysuch that andAlso, we have the affine transformation defined bysuch that andHence, setting , we have andFollowing Property 2, we have

Theorem 4. Let and be curves and a point in . We suppose that is a -fold point of and an -fold point of , respectively. If , then there is a curve such thatwhere is an -fold point of and .

Proof. When , that is, is, respectively, a 1-fold point of and a 1-fold point of . By Definition 6, we know that there exists a line such thatFollowing Lemma 6, there exists an affine transformation defined bysuch thatwhere , , , and . Hence, we can writefor some polynomials and in with and . Also, we can writefor some polynomials and in with and . From Property 2, we havewhere with and is an -fold point of with .
By (48) and the definition of , we know that there exists an affine transformation defined bysuch that , , and . Therefore, according to the fact that , we havewhere is an -fold point of with . Thus, we obtain the assertion when .
When , that is, is, respectively, a -fold point of and an -fold point of . Following Definition 6, we know that there exists a line such thatBy Lemma 6 there exists an affine transformation defined bysuch thatwhere , , and . From Lemma 5, we have that is a -fold point of if and only if , which means that is not a factor of . This tells that the coefficient of the term in is nonzero, and thenwe can writewhere and are polynomials in with and . Similarly, we can writewhere and are polynomials in in with and . Thus, according to (115) and (116), we havewhere with and . On the other hand, following (117), we also have thatwhere . Therefore, we haveIt follows thatHence, by Property 2, we have thatSince , following Lemma 5, we have that is an -fold point of . By (48) and the definition of , we know that there exists an affine transformation defined bysuch that , , and . Therefore, according to the fact that , we havewhere is an -fold point of and . Thus, we obtain the assertion.
Hartshorne (([7], Chapter 1)) proved that the intersection multiplicity of distinct curves at a point in is not less than . Similar as the proof of Theorem 2, we also have the following corollaries by using the fold point.

Corollary 2. Let and be curves and a point in . Suppose that is a -fold point of and an -fold point of , respectively, then

Corollary 3. Let and be curves in . We assume that the origin is a -fold point of and an -fold point of , respectively. Let (.,) be the sum of the terms of the degree (.,) in (.,). Then, if and only if and have a common factor of positive degrees. Equivalently, if and only if and have no common factors of positive degrees.

Example 4. We calculate the intersection multiplicity at the origin of two algebraic curves and in .
It is obvious that and . From Definition 6, we know that the origin is a 2-fold point of both and . By Corollary 2, we know that . Since and have no common factors, we have following Corollary 3.
In fact, we havewhere and , which implies thatThen,Thus, following Property 2 we haveNote that the origin is a 1-fold point of . Thus, from Theorem 4, we can also obtain that . Similarly, we havewhere and , which implies thatThis means thatFollowing Proposition 3.5, since , we have andTherefore, we have .

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The first author acknowledges the support by the Innovation Research for the Postgraduates of Guangzhou University (2021GDJC-M12), and the second author acknowledges the support by the National Natural Science Foundation of China (11701111; 12031003) and the Ministry of Science and Technology of China (CSN-XXZ-20120700649).