On Intersection Multiplicity of Algebraic Curves

(Shandong Vocational College of Science and Technology,Weifang 261053,China)

Abstract:In this paper,we study the intersection multiplicity of algebraic curves at a point both in R2and in real projective plane P2.We introduce the fold point of curves and provide conditions for the relations between the intersection multiplicity of curves at a point and the folds of the point.

Key words:Algebraic curves;intersection multiplicity;projective transformation;d-fold point

§1. Introduction

Let F(x,y,z)=0 and G(x,y,z)=0 be two algebraic curves and P a point in P2.It is known that the intersection multiplicity of curves F and G at P is the number of times that the curves F=0 and G=0 intersect at P(cf.[4]).In fact,there are di ff erent ways to define the intersection multiplicity of algebraic curves at a point(cf.[3],[5]-[6],[8]).In algebraic geometry,an important way to study algebraic curves is to analyze its intersections with other curves,and the intersection multiplicity is a useful tool to describe their intersections.Therefore,there is a natural question to ask how to determine the intersection multiplicity of algebraic curves.R.Walker(cf.[7])determined the intersection multiplicity by means of formal power series and the resultant of two algebraic curves,and proved that if P is a point of a curve F=0(resp.G=0)with multiplicity r(resp.s),then F and G intersect at P at least rs times,and exactly rs times when the curves F and G do not have common tangents at P.On the other hand,G.S.Avagyan determined the intersection multiplicity by means of operators with partial derivatives,and proved the same result as the above(cf.[1]-[2]).However,in general,it is difficult to identify the intersection multiplicity of algebraic curves.In this paper,we determine the intersection multiplicity of algebraic curves at a point by using the folds of the point,and show the Theorem3.4 in section 3.

This paper is organized as follows.In section 2,we review some fundamental facts about the intersection multiplicity of curves at a point in R2and P2.In section 3,we introduce the fold point of curves and provide conditions for the relations between the intersection multiplicity of curves at a point and the folds of the point.

§2. Preliminaries

In this section,we introduce the intersection multiplicity of algebraic curves at a point respective in P2and in the real projective plane P2,i.e.,(R3?{0})/～,where the equivalence relation～is defined by(x,y,z)～(x0,y0,z0)if there exists a nonzero λ∈R,such that(x0,y0,z0)=(λx,λy,λz).We also introduce projective transformations,which are linear changes of coordinates and preserve the intersection multiplicity of curves.

2.1 Intersection multiplicity of curves in R2

Let f(x,y)=0 and g(x,y)=0 be curves,and let P be any point in R2.We denote the intersection multiplicity of f and g at P by IP(f,g).Since the definition is unintuitive,we first list several intersection properties of curves(cf.[4],[8]).

Property 2.1[4,Section1]Let f(x,y)=0,g(x,y)=0 and h(x,y)=0 be curves,and P a point in R2.Then

1.IP(f,g)≥0,and IP(f,g)≥1 if and only if f=0 and g=0 both contain the point P.

2.IP(f,g)=IP(f,g+fh).

3.IP(f,gh)=IP(f,g)+IP(f,h).

4.IP(f,gh)=IP(f,g)if h does not contain the point P.

Let f(x,y)be a non-zero polynomial.Then the order of f at(0,0)is said to be the smallest degree of the terms of f,denoted by ord(f).Evidently,ord(fg)=ord(f)+ord(g)for polynomials f,g.Let o denotes the origin(0,0)in R2.

Lemma 2.2[4,Theorem1.11]Let y=p(x)and g(x,y)=0 be curves.Assume that y=p(x)contains the origin o and that y?p(x)is not a factor of g(x,y).Then

Corollary 2.3 Let f(x,y)=0 be a curve that contains the origin o,and let fd(x,y)be the sum of the terms of degree d in f(x,y),where d=ord(f(x,y)).Assume that l=0 is a line through the origin o.Then Io(l,f)>d if l is a factor of fdand Io(l,f)=d if l is not a factor of fd.

2.2 Intersection multiplicity of curves in P2

We extend the intersection multiplicity of curves from R2to P2.Let F(x,y,z)=0 and G(x,y,z)=0 be curves and P a point in P2.Similar as in Section 2.1,we denote the intersection multiplicity of F and G at P by IP(F,G).

Property 2.4[4,Theorem3.6]In P2,let F(x,y,z)=0,G(x,y,z)=0 and H(x,y,z)=0 be curves,and let P be a point.Then

1.IP(F,G)≥0,and IP(F,G)≥1 if and only if F and G both contain P.

2.IP(F,G)=IP(G,F).

If we set f(x,y)=F(x,y,1),then a point(x,y)of R2which lies on the curve f=0 if and only if the corresponding point(x,y,1)lies on the curve F=0.Let g(x,y)=G(x,y,1).Then the number of intersection points of f=0 and g=0 is the number of points of the form(x,y,1)lying on both F=0 and G=0.Let O be the origin(0,0,1)in P2.

Lemma 2.5[4,Property3.1]Let F(x,y,z)=0 and G(x,y,z)=0 be curves in P2,and let f(x,y)=F(x,y,1)and g(x,y)=G(x,y,1).Then

In projective geometry,projective transformations play a central role.A projective transformation of P2is a map T:P2→P2defined by T:(x,y,z)7→A(x,y,z),where A∈GL(3,R)is an invertible 3×3 matrix,and it is always a bijection which maps lines to lines.Note that if T is a projective transformation and points P,Q lie on a line L,then T(P)and T(Q)lie on T(L).Similarly,if lines L1and L2intersect at a point P0,then the lines T(L1)and T(L2)intersect at the point T(P0).In fact,we can transform any four points,no three of which are collinear,into any four points,no three of which are collinear.

Lemma 2.6[4,Theorem3.4]In P2,let A,B,C,D,be four points,no three of which are collinear,and let A0,B0,C0,D0,be four points,no three of which are collinear.Then there is a projective transformation that maps A,B,C,D to A0,B0,C0,D0,respectively.

Since projective transformations preserve intersection multiplicity of curves(cf.[4]),we canfind the number of times that two curves intersect at any point in P2.

Lemma 2.7[4,Property3.5]Let F(x,y,z)=0 and G(x,y,z)=0 be curves in P2,and let T be a projective transformation that maps(x,y,z)to(x0,y0,z0)in P2.Assume that F0(x0,y0,z0)and G0(x0,y0,z0)are the images of F(x,y,z)and G(x,y,z)under T,respectively.Then

for any point P∈P2,where P0=T(P).

§3. Fold Point and Intersection Multiplicity of Curves

In this section,we introduce the fold point of curves and provide conditions for the relations between the intersection multiplicity of curves at a point and the folds of the point in R2and in P2.

Definition 3.1 Let G(x,y,z)=0 be a curve and P a point in P2.Then we call P a d-fold point of G=0 if there is a non-negative integer d,such that all but a finite number of lines intersect G exactly d times at P,and all other lines intersect G more than d times at P and there are at most d such lines.

Similarly,we can define a e-fold point of a curve in R2for a non-negative integer e.

Lemma 3.2 Let F(x,y,z)=0 be a curve that contains a point P in P2,and let d be a non-negative integer.Set f(x,y)=F(x,y,1).Then the point P is a d-fold point of F=0 if and only if ord(f)=d.

Proof By Lemma 2.6,there is a projective transformation T that maps P to the origin O=(0,0,1).Following Lemma 2.5 and Lemma 2.7,it is suffice to show that the origin o is a d-fold point of f=0 if and only if ord(f)=d.Let l=px+qy=0 be a line in R2,where p,q are not all zero.We may assume that q 6=0.

Suppose that the origin o is a d-fold point of f=0 and that ord(f)=d0.Let fd0be the sum of the terms of degree d0in f.From Lemma 2.5,we know that if l is a factor of fd0,then Io(l,f)=ord(f(x,?))>d0;and if l is not a factor of fd0,then Io(l,f)=ord(f(x,?))=d0.From Definition 3.1,we obtain that d0=d.

Conversely,assume that ord(f)=d.Let fdbe the sum of the terms of degree d in f.By

Lemma 2.2,if l is a factor of fd,then Io(l,f)=ord(f(x,?))>d;and if l is not a factor of fd,then Io(l,f)=ord(f(x,?))=d.This means that the origin o is a d-fold point of f=0 following Definition 3.1.

Let d,e,k be non-negative integers.

Theorem 3.3 Let F(x,y,z)=0 and G(x,y,z)=0 be curves that intersect at a point P in P2.Assume that P is a d-fold point of F=0 and a e-fold point of G=0.If 1≤d≤e,then there is a curve H=0 such that

where P is a k-fold point of H=0 for k≥e?1.

Proof Since P is a d-fold point of F=0,there exists a line L=0 through the point P,such that IP(L,F)=d.By Lemma 2.6,we may assume that P=(0,0,1)and L=y=0.Let f(x,y)=F(x,y,1)and let fdbe the sum of the terms of degree d in f.Then we can factor for distinct lines pix+qiy=0,where siare positive integers and r(x,y)is a polynomial that has no factor of degree 1.Following Lemma 2.5,we have that IP(F,y)=Io(f,y)=d,which implies that y is not a factor of fdby Corollary 2.3.In other words,pi6=0 for all i and the coefficient of the term xd?(s1+···+sk)in r is non-zero.Thus,the coefficient of the term xdin fdis non-zero,and we can write

for some polynomials p(x)with p(0)6=0 and s(x,y),every term of s has degree at least d?1.Set g(x,y)=G(x,y,1).Similarly,we can write

for some polynomials q(x)and t(x,y),every term of t has degree at least e?1.Following Lemma 2.5 and Property 2.1(2),(3)and(4),we have

Since ord(s)≥d?1 and ord(t)≥e?1,we can write

for some polynomials m(x)and n(x,y),every term of n has degree at least d?2,and

for some polynomials u(x)and v(x,y),every term of v has degree at least e?2.Set h=tp? sxe?dq.Then

which has order at least e?1,and the origin o is a k-fold point of h=0 for k≥e?1 following Lemma 3.2.Set H(x,y,1)=h(x,y)and multiply each term of h by the power of z to produce a term of degree deg(h),where deg(h)is the maximal degree of the terms of h.Thus,we obtain a homogeneous polynomial H(x,y,z)which satisfies IP(F,H)=Io(f,h)from Lemma 2.5.By Lemma 3.2,we know that P is a k-fold point of H=0.Hence,we obtain the assertion.

Theorem 3.4 Let F(x,y,z)=0 and G(x,y,z)=0 be curves and P a point in P2.Assume that P is a d-fold point of F=0 and a e-fold point of G=0,then

Proof We may assume that d≤e by Property 2.4(2).When d=1 and e=1,we have that IP(F,G)≥1 following Property 2.4(1).Fix the integer e.

Assume that d=1.From Theorem 3.3,there exists a curve H1=0,such that

and P is a k1-fold point of H1=0 for k1≥e?1.If k1=1,then IP(F,G)≥2≥de with e≤2.If k1>1,then there exists a curve H2=0,such that

and P is a k2-fold point of H2=0 for k2≥k1?1≥e?2.Assume that there exists a curve Hm=0 for some positive integer m with m

and P is a km-fold point of Hm=0 for km≥km?1?1≥ e?m.If km=1,then IP(F,G)≥m+1≥de with e≤m+1.If km>1,then there exists a curve Hm+1=0,such that

and P is a km+1-fold point of Hm+1=0 for km+1≥km?1≥e?(m+1).Inductively,if m=e?1 and ke?1=1,then IP(F,G)≥ (e?1)+1=e=de.If ke?1>1,then there exists a curve He=0,such that

and P is a ke-fold point of He=0 for ke≥ ke?1?1≥ 0.Clearly,IP(F,G)≥ de.Hence,we obtain that IP(F,G)≥de when d=1.

Assume that IP(F,G)≥le if P is a l-fold point of F=0 for 1≤l

with e≤l+1 by the assumption and Property 2.4(2).If>l,then there exists a curve=0,such that

Therefore,by induction and the arbitrary of e,we obtain the assertion.

Corollary 3.5 Let f(x,y)=0 and g(x,y)=0 be curves in R2.Assume that the origin o is a d-fold point of f=0 and a e-fold point of g=0.Then Io(f,g)≥de.

Proof Let deg(f)(resp.deg(g))denote the maximal degree of the terms of f(resp.g).Multiply each term of f(resp.g)by the power of z to produce a term of degree deg(f)(resp.deg(g)),we obtain a homogeneous polynomial and denoted F(x,y,z)(resp.G(x,y,z)).From Lemma 3.2,we know that the origin O=(0,0,1)is a d-fold point of F=0 and a e-fold point of G=0.By Lemma 2.5 and Theorem 3.4,we have that IO(F,G)=Io(f,g)≥de.Hence,the assertion holds.