## Fixing Bezout’s Theorem

Bezout’s theorem in algebraic geometry is one of those simple facts that manages to capture the heart and style of its field.  It states that any two irreducible curves $C_1$ and $C_2$ in $\mathbb{C}^2$ usually intersect in $deg(C_1)\cdot deg(C_2)$ points (where the degree of a curve is the degree of the polynomial that defines it).

Now, very few mathematicians will stand for a ‘usually’ in their theorems, and the most basic form of Bezout’s theorem is typically stated differently – so as to be a real theorem.  However, my favorite aspect of the theorem is that figuring out how to fix the ‘usually’ has repeatedly foreshadowed the development of algebraic geometry as a whole.

In its simplest cases, Bezout’s theorem is familiar even to most math undergrads.  For, if $C_1$ is a degree 1 curve, then it is a line.  Restricting the defining equation for $C_2$ to this line gives a polynomial in one variable of (usually) the same degree as $C_2$, and so by the Fundamental Theorem of Algebra, it (usually) has $deg(C_2)$ points.  I have glossed over the fact that restricting to the line, which amounts to eliminating a variable, can sometimes cause highest degree terms to vanish; and also that sometimes two roots of a polynomial will coincide.  Still, the underlying intuition is the Fundamental Theorem of Algebra, which everyone should find reassuring and familiar.

For higher degrees, the proof is less immediate, but no less intuitive.  The two curves always intersect at exactly the proscribed number of points – unless some numbers happen to cancel/coincide.

Geometrically, what is going wrong?  There are two kinds of behavior that are causing the theorem to be break down(as I have stated it).

The first is most easily seen in the case of two parallel lines.  They both have degree 1, and so Bezout says they should intersect, but Euclid and reality disagree.  Higher degree curves can exhibit similar behavior, such as when parallel asymptotes cause an intersection point to be lost.

The second problem is when the two curves intersect too well at a given point of intersection.  As a simple example, take $C_1$ to be the curve defined by $y=0$ (also know as the x-axis), and let $C_2$ be defined by $y=x^n$ (also known as the graph of $x^n$).  These curves do intersect at $(0,0)$, but this is the only place they intersect.  This intersection point has multiplicity; that is, it wants to correspond to more than one point ($n$ in this case), but there is no room in the geometric viewpoint to count the same point more than once.

First Fix: Projective Geometry

The first problem is pretty easy to fix.  As you might have guessed, the answer is to think of two parallel lines as ‘intersecting at infinity’.  $\mathbb{C}^2$ can be compactified by adjoining a sphere at infinity.  A path leaving $\mathbb{C}^2$ has a limit if it has an asymptote.  This new space we have constructed is called $\mathbb{C}P^2$, and it is the second member of the family $\mathbb{C}P^n$ of complex projective spaces (the first member is the Riemann sphere).

If we think of our curves $C_1$ and $C_2$ as sitting inside of $\mathbb{C}P^2$, then there is a canonical way to extend them to the sphere at infinity.  Adding these points at infinity will magically add points of intersection that make Bezout’s theorem (closer to being) true!  For example, our two parallel lines now have exactly one intersection point, as we hoped would be true.

This is a basic example of ‘Projective Geometry’, which is the study of varieties not in $\mathbb{C}^n$, but their compactifications in $\mathbb{C}P^n$.  Theorems tend to be cleaner here, and in general this is seems to be the more natural home for varieties.

Second Fix: Intersection Theory

I claim that projective geometry fixed the problem of points at infinity, so the only remaining problem is that figuring out how to count points with the right multiplicity.

The most basic solution is to notice that any time two curves intersect with multiplicity $n>1$, you can slide one of the curves a bit in some direction to split the bad point into n points.  Thus, Bezout’s theorem is always true, up to an infinitesmal slide.  This has the advantage of being conceptually simple, but its not very useful for computations.

We can also develop a rigorous theory for counting intersection multiplicities, appropriately called intersection theory.  The details are a bit more technical than I am aiming for with this post, but the idea is that one declares two curves ‘the same’ if you can find an analytic function on the compliment of the first curve that vanishes only on the second curve.  This rigid relationship is called ‘linear equivalence’, and its a more algebraic version of the above sliding intuition.  The number of intersection points of two curves depends only the linear-equivalence class… except for a small number of exceptions.  Thus, we define the intersection number of two curves to be the number of intersections of any two ‘generic’ curves linearly-equivalent to the original pair.

This is mostly just a fancy way of declaring that its ok to slide a curve a small amount in order to resolve a point of multiplicity.  However, this is a more effective computational tool, and intersection theory has some nice benefits.  For example, intersection theory can be robustly generalized to higher dimensional varieties.  Instead of a $\mathbb{Z}$-valued inner-product like we just constructed, we get a ring called the Chow ring.

Second Fix, Take Two: Schemes

There is a slightly different take on this that isn’t explicitly necessary, but I like since it points us in the direction of another important evolution in algebraic geometry.  The above fix was secretly just a weakening of Bezout’s theorem, from talking about curves intersecting, to talking about equivalence classes of curves intersecting.

Let us instead declare that the deficiency was not with the theorem, but with our notion of geometry instead.  Perhaps our definition of ‘point’ was too primitive to distinguish between a simple intersection, and one of higher multiplicity.  These are bold statements, and they require a bold theory to pull off; but the theory of schemes is just bold enough.

Very crudely, a scheme in this context behaves like a variety, with a distinguished sheaf on it that dictates what the ‘ring of rational functions’ looks like over any (good) open set.  With the extra data provided by this sheaf, we can distinguished between two schemes whose underlying variety is the same.

Take for example, a point… a sheaf on a point is the same as a its ring of global sections.  This ring can’t be anything; it must be a commutative $\mathbb{C}$ algebra such that modding out by the Jacobson radical gives $\mathbb{C}$.  This doesn’t leave room for too much, but we still get multiple different schemes that look like points to the naked eye (ie, as a variety).

Now, there is also a way to intersect schemes, so we can see what happens if we intersect $C_1=\{y=0\}$ and $C_2=\{y=x^2\}$.  We get a point whose global sections looks like $\mathbb{C}[\epsilon]/\epsilon^2$ (which readers of previous posts might recall is one of my favoritest rings).  Ah ha!  We get a point of a totally different flavor than what we would have gotten if we’d looked at a simple intersection (its global sections would have been $\mathbb{C}$).  By assigning to each distinct flavor of point a ‘multiplicity’ equal to its dimension as a $\mathbb{C}$ vector space, we can again make Bezout’s Theorem work.

Third Fix…? : Derived Schemes

But wait, I said there were only two things wrong with Bezout’s theorem – what else is there to fix?  Well, I have stealthily concealed a third minor error in the statement, to see if readers would ignore it unthinkingly.  The trick is that I never forbid that the two curves coincided.  Its instinct to dismiss such cases out of hand, since the ‘number’ of intersection points doesn’t make sense.

Perhaps we shouldn’t be so hasty; after all, the other two fixes involved building techniques that turned out to be useful for wholly unrelated reasons.  According to Jacob Lurie’s engrossing GRASP lecture, this error can also be fixed, by passing to an even richer version of geometry, known as derived algebraic geometry.  I should qualify the following by saying that I know almost nothing about the subject, and so I am parroting cool ideas I have heard others express.

A curve $C_1$ in $\mathbb{C}^2$ is the zero-set of some polynomial $f$.  Given two polynomials on $\mathbb{C}^2$, their restriction to $C_1$ is the same only if they differ by some multiple of $f$.  Thus, the ring of polynomial functions on $C_1$ looks like $\mathbb{C}[x,y]/f$.  If I had another curve $C_2$, it would be defined by a polynomial $g$, and the ring correspond to the scheme of the intersections of $C_1$ and $C_2$ is $\mathbb{C}[x,y]/\{f,g\}$.  This breaks down if $f=g$, since quotienting by $f$ twice is the same as quotienting by it once.

To fix this, let us first replace rings with topological rings.  In practice, topological rings aren’t the right idea to work with, but they will suffice for conveying intuition.  A ring then becomes a set of discrete points, with the appropriate ring axioms, etc.  We can now think of the act of quotienting by $f$ as connecting two points by a line every time a multiple of $f$ takes one to the other.  We should also add triangles and higher simplices between appropriate compositions.  If $f$ is nice enough (cancelable),  we get a set of contractable pieces, each of which correspond to an element of the quotient.  Since we are trying to think of topological rings only up to homotopy, this gives us the old, boring notion of the quotient.

However, if $f$ wasn’t nice, then we added some non-trivial loops.  The connected components might still be isomorphic to the boring quotient, but suddenly non-trivial first homology has emerged.  We can define an Euler characteristic as usual, the alternating sum of dimensions of the homology.  Now, if I have two curves that coincide, I can say that their intersection number is the Euler characteristic of the corresponding topological ring, and I can ask if this is equal to the product of the degrees.  I believe this is true, though a quick shuffling through my references hasn’t yielded a confirmation.

Derived algebraic geometry is an exciting field that I would like to learn more about.  As near as I can tell, it is an attempt to bring the idea of homotopy equivalence into the core of scheme theory, with the goal of explaining such phenomena as stacks that really should have a tangent space that isn’t a vector space, but a complex of vector spaces (up to homotopy).  If you are interested in learning more, I would recommend the above GRASP lecture, the (rather long) papers at Lurie’s homepage, and some of Toen’s lecture notes online

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### 29 Responses to “Fixing Bezout’s Theorem”

A very nice piece on the Bezout Theorem. I just have on nit picking remark that it’s the Jacobson radical, not the Jacobsen radical, you want to mod out by!

2. Mikael Johansson Says:

Mwah! THAT’s what derived algebraic geometry is about? *droooooooool*

I just got my “Have to learn this”-list expanded. By a couple of pages.

3. Anonymous Says:

Fundamental Theorem of Calculus Fundamental Theorem of Algebra

4. Greg Muller Says:

Whoops… thanks for the corrections. I’ve fixed them above.

5. A.J. Tolland Says:

an attempt to bring the idea of homotopy equivalence into the core of scheme theory

You can turn this around, too, and say that the point of derived algebraic geometry is that all the usual algebraic geometry ideas — sheaves, functors of points, stacks, and topoi — seem to work just fine if, instead of commutative rings, we use other objects that behave like commutative rings. Simplicial commutative rings, E_infinity ring spectra, commutative DGAs, etc,… So derived algebraic geometry is also about importing ideas from scheme theory into homotopy theory.

6. Drew Armstrong Says:

I think you need to check the definition of “proscribed”.

7. Algebraic Geometry - A Historical Sketch I « Rigorous Trivialities Says:

[…] curves will always be . This theorem is nifty in other ways, and to see one of them check out this post at the Everything […]

8. Bezout’s Theorem « Rigorous Trivialities Says:

[…] Now, if we specify that we’re living in the projective plane, then we take be distinct curves of degrees and , that is, irreducible one dimensional projective varieties who intersect in a finite collection of points , we have that . This is because points have Hilbert polynomial 1. (If we don’t require that they only intersect in a finite collection of points, things get trickier. However, a subject called Derived Algebraic Geometry appears to help understand this case, and you can read a bit about that here.) […]

9. carey rosenthal Says:

I’m no mathematician, but here is a simple question suppose you had 2 quadratics in x and y.. Bezout’s theorem predicts 4 solutions. fine. But suppose you get the Grobner basis for this set of 2 polynomials (in the plex form). So b1 is a 4th order polynomial in x and b2 is lst order in y and 3rd order in x. If you apply Bezout’s theorem to this set of 2 polynomials the number of intersections seems to have risen to 4×3=12..What’s happening? Do b1 and b2 have the other 8 solutions at infinity..are there multiple roots. I don’t get it!

10. Charles Siegel Says:

Carey,

That can’t be all there is to the Groebner basis, because it won’t generate the same ideal. So there must be more polynomials, which would cut down the number of solutions further. Chosen at random, they’d cut it down to zero, but they aren’t random at all, so it works out.

Also, you’d expect 16, because both of the two polynomials are total degree 4.

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18. ceasesiu Says:

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APPLICATION OF BEZOUT THEOREM
ABSTRACT
It has frequently been pointed out that, until the time when CANTOR championed the actual infinite, it had been traditionally rejected by all mathematicians─a tradition that is traced back to ARISTOTLE, and which counts GAUSS and CAUCHY as 19th century examples. In this section, however, we will see that this generalization does not hold water against the views held by RIEMANN and DEDEKIND from the 1850s.

19. mociociu Says:

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20. nasanu Says:

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21. manceanu Says:

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22. podeaciu Says:

It is clear that the system has a single equilibrium point at the origin for all r. The Jacobian matrix is given by

and , and hence the equilibrium point is an attractor if and a repeller if where .

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23. ceanegiu Says:

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In closing, I would like to thank two people who assisted me in writing this paper. First, I thank my brother, Matt Black, who spent several hours helping me visualize the surface area problem and suggested tipping the triangle onto the plane to derive the formulas. Second, I thank Dr. Guthrie who helped me wrestle with an attempt to integrate the spherical triangle region. Though we were unsuccessful in finding the surface area this way his help on the use of the vectors was much appreciated.

24. saianu Says:

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25. cimciolu Says:

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• Poles

In the Laurent expansion

f(z) = an (z – z0)n,

If an = 0 for n < -m < 0 and a-m  0, we say that z0 is a pole of order m. For instance, if m = 1, that is a-1/(z-z0) is the first non-vanishing term, we have a pole of order one, often called a simple pole.

If, on the other hand, the summation continues to n = – , the z0 is a pole of infinite order and is called an essential singularity. One point of fundamental difference between a pole of finite order and an essential singularity is that a pole of order m can be removed by multiplying f(z) by (z-z0)m. This obviously cannot be done for an essential singularity.

The behavior of f(z) as z   is defined in terms of the behavior of f(1/t) as t  0. Consider the function

As z  , we replace the z by 1/t to obtain

Clearly, from the definition, sin z has an essential singularity at . This result could be expected from the following analysis.

When x = 0, sin z = sin iy = = i sinh y,
which approaches  exponentially as y  . Thus, the absolute value of sin z is not bounded.

• Branch points
There is another sort of singularity. Consider

f(z) = z a,
in which a is not an integer, As z moves around the unit circle from e 0i to e 2i,

f(z)  e 2ai  e 0i.

We have a branch point at the origin and another at infinity. The points e 0i and e 2i in the z -plane coincide but they lead to different values of f(z).

The problem is resolved by constructing a cut line joining both branch points so that f(z) will be uniquely specified for a given point in the z-plane.

Note that a function with a branch point and a required cut line will not be continuous across the cut line. In general, there will be a phase difference on opposite sides of this cut line. Hence line integrals on opposite sides of this cut line will not generally cancel each other.

The contour line used to convert a multiply connected region into a simply connected region (Section 1.3) is completely different. The function is continuous across this contour line, and no phase difference exists.

Example:

Consider

f(z) = (z2 – 1) 1/2 = (z + 1)1/2(z – 1)1/2

The first factor on the right hand side(RHS), (z+1)1/2, has a branch point at z = -1. The second factor has one at z = 1. At infinity f(z) has a simple pole. The cut line has to connect both branch points. To check on the possibility of taking the line segment jointing z = +1 and -1 as a cut line, let us follow the phases of these two factors as we move along the contour shown in Fig2.1.

Fig.2.1

For convenience, let z + 1 =r e i and z – 1 =  e i. Then the phase of f(z) is ()/2. At point 1,  = 0; 1 to 2, , but  – unchanged; then  stays constant until 6; 6 to 7, = .  increases by 2 as we move from 3 to 5. The phase of f(z) is tabulated in the final column of table2.1.

Table2.1 Phase Angle

(1). The phase at points 5 and 6 is not the same as the phase at 2 and 3. This behavior can be expected at a branch point cut line.
(2). The phase at 7 exceeds that at 1 by 2 and f(z) = (z -1)1/2 is therefore single-valued for the contour shown, encircling both branch points.

If we take the x-axis -1 ≤ x ≤ 1 as a cut line, f(z) is uniquely specified. Alternatively, the positive x-

26. ghepingu Says:

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.
Lagrange’s Equations for Conservative Systems
In a conservative system, the generalised forces can be derived from a potential function V. Potential functions can represent the effects of:
• Ideal springs (energy storage) – Energy storage =
• Gravity – energy storage =
When dealing with such systems, a so-called ‘Lagrangian’ L is defined as follows:

(2)
The appropriate form of Lagrange’s equations for conservative systems is given by:

(3)
Application to a 2-degree-of-freedom system
This is a conservative system, and so we can use form (3) of the equation above. The springs have negligible mass, and there is no energy loss in system. We define generalised co-ordinates x (displacement from equilibrium – this eliminates any gravity forces) and , the angular displacement of the pulley whose moment of inertia through its axis is I.

Kinetic and power and mention Bezout Theorem for other complex function

27. ciomogea Says:

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28. ciunaciu Says:

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29. ducudu Says:

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