UTM曲线概论：解析代数曲线的直观入门

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"《UTM Conics and Cubics: A Concrete Introduction to Algebraic Curves》是一本专为本科生设计的数学教材，由Robert Bix撰写，书号为038731802X。本书将焦点放在了二次曲线（conics）和三次曲线（cubics）上，提供了一种直观且实践性强的方法来理解代数曲线的概念。在代数几何领域，这些曲线是基本构造，它们在解析几何、项目ive几何以及其他数学分支中有广泛的应用。 UTM系列（Undergraduate Texts in Mathematics）是由S.Axler和K.A.Ribet共同编辑的一套教材，旨在为大学低年级学生提供深入浅出的数学学习资源。《UTM Conics and Cubics》在该系列中占据一席之地，它不仅介绍了基本的二次方程（如椭圆、双曲线和抛物线）和三次方程的性质，还通过实际问题和几何直观，帮助读者建立起对这些曲线几何特性的深刻认识。在阅读过程中，学生可以了解到如何通过方程形式来描述这些曲线，如何分析它们的轨迹、对称性以及与其他代数对象的关联。此外，书中可能包含解方程、求极值点、研究曲线的接触性质等实际操作技巧，这些都是代数曲线理论的重要组成部分。相比于抽象的理论推导，本书更注重于提供一种“concrete introduction”，即通过具体实例和应用来引导读者探索数学之美。这使得对于初学者来说，这本书不仅是学术资料，也是理解和掌握代数曲线概念的有效工具。同时，作者可能会在书中引用或比较其他经典教材，如Understanding Analysis（Abbott）、Mathematics: A Concise History and Philosophy（Anglin）、Introduction to Analytic Number Theory（Apostol）等，以展现代数曲线的历史背景和发展脉络，帮助读者建立全面的数学视野。《UTM Conics and Cubics》是一本实用而深入的入门教材，适合那些希望系统学习代数曲线的学生，无论是在预备课程中还是作为独立的学习参考资料，都能从中受益匪浅。"

We use the term intersection properties to refer to Properties 1.1–1.6

and further properties introduced in Sections 3, 11, and 12. We must

prove that we can assign values I

(f, g) for all pairs of curves f and g so

that Properties 1.1–1.6 hold. We postpone this proof until Chapter IV so

that we can proceed with our main task, using intersection properties to

study curves. Of course, the results we obtain depend on our proving the

intersection properties in Chapter IV.

In the rest of this section, we show how Properties 1.1–1.6 can be used

to comput e the intersection multiplicity of two curves at the origin. The

discussion accompanying Figures 1.1–1.3 suggests that I

(f, g) measures

how closely the curves f and g approach each other at the origin. When

f is a factor of g, the graph of g ¼ 0 contains the graph of f ¼ 0. Thus, we

are led to expect the following resu lt:

Theorem 1.7

If f and g are polynomials such that f is a factor of g and the curve f ¼ 0

contains the origin O, then I

(f, g) is y.

Proof

Consider ﬁrst the case where g is the zero polynomial 0. (The theorem

includes this case because the zero polynomial has every polynomial f

as a factor, since 0 ¼ f 0.) Since I

(f,0)V 1 (by Property 1.3), it follo ws

for every positive integer n that

n U nI

(f,0) ¼ I

(f,0

) (by Property 1:6)

¼ I

(f,0):

Because this holds for every positive integer n, I

(f, 0) must be y.

In general, if g is any polynomial that has f as a factor, we can write

g ¼ fh for a polynomial h. Then we have

(f, g) ¼ I

(f, fh)

¼ I

(f, fh fh) (by Property 1.5)

¼ I

(f,0) ¼y,

by the previous paragraph. r

The proof of Theorem 1.7 shows why we needed to allow inﬁnite

intersection multiplicities in Property 1.1.

The foll owing result shows that we can disregard factors that do not

contain the origin when we compute intersection multiplicities at the

origin:

Theorem 1.8

If f , g, and h are curves and g does not contain the origin, we have

(f, gh) ¼ I

(f, h):

8 I. Intersections of Curves

Of course, a polynomial p(x) in one variable x is a ﬁnite sum of terms

of the form ex

, where e is a real number and i is a nonnegative integer.

By generalizing the previous paragraph, we can ﬁnd the number of

times that a curve of the form y ¼ p(x) intersects any curve g(x, y) ¼ 0

at the origin. This is easy to do because we do not need long division to

ﬁnd the remainder when g(x, y) is divided by y  p(x) with respect to y.

The next theorem shows that the remainder is g(x, p(x)), the result of

substituting p(x) for y in g(x, y). For example, we did not have to use

long divisio n above to ﬁnd the remainder when y

þ 2xy þx

is divided

by y  x

. All we needed to do was substitute x

for y in y

þ 2xy þx

ﬁnd that the remainder is (x

)

þ 2x(x

) þ x

¼ 2x

þ 2x

, as before.

Theorem 1.9

Let p(x) and g(x, y) be polynomials.

(i) If we use long division with respect to y to divide g(x, y) by y  p(x), the

remainder is g(x, p(x)). This means that there is a polynomial h(x, y)

such that

g(x, y) ¼ (y  p(x))h(x, y) þg(x, p(x)): (4)

(ii) In particular, y  p(x) is a factor of g(x, y) if and only if g(x, p(x)) is the

zero polynomial.

Proof

(i) Let h(x, y) be the quotient when we use long division with respect to y

to divide y  p(x) into g(x, y). The remainder is a polynomial r(x)inx be-

cause each step of the division eliminates the highest remaining power

of y. We have

g(x, y) ¼ (y  p(x))h(x, y) þr(x): (5)

Substituting p(x) for y in (5) makes y  p(x) zero and shows that

g(x, p(x)) ¼ r(x):

Together with (5), this gives (4).

(ii) If g(x, p(x)) is the zero polynomial, (4) shows that y  p(x)isa

factor of g(x, y). Conversely, if y  p(x) is a factor of g(x, y), we can write

g(x, y) ¼ (y  p(x))k(x, y)

for a polynomial k(x, y). Substituting p(x) for y shows that g(x, p(x)) is

zero. r

We obtain a familiar result from Theorem 1.9 if we assume that x does

not appear in p or g. Then p is a real number b, and g is a polynomial

g(y)iny. When we divide g(y)byy  b, the quotient is a polynomial

10 I. Intersections of Curves

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