{Bézout’s Theorem

Philosophy of Mathematics

Bézout’s Theorem is a statement about how curves cross each other.

In its modern form it’s usually generalized to many dimensions but we’ll just consider the simplest case of plane curves.

We’re looking at it because: It’s a famous theorem; It builds on some ideas from last time; It requires a considerable amount of care

to get it to work, and you may feel some of this crosses the line into “cheating”.

Introduction

Two words of warning in fact:

1. This is the most maths we’ll do in a session. It’s going to be a bit vague and hand-waving. Try to get a sense of the general ideas without getting distracted by specific details.

2. In order to bring out the philosophical issues I’m going to present this in a way that’s a bit sensational and rather unfair to the mathematical world.

Word of Warning

Many important problems in maths can be reduced to solving a system of equations, which means finding the points where the curves they represent intersect.

It’s useful at the start to know how many solutions we’re looking for – this is the question Bézout's theorem answers.

Motivation

Intersections of curves were studied in antiquity. The problem of doubling the cube was attacked in this way, which led to the discovery of conics.

A simpler version of Bézout's theorem was used by Isaac Newton in his proof of 1.VI.28 in the Principia (1687).

Étienne Bézout published a more general version in 1779.

Jean-Victor Poncelet (c1822) made important observations about the theorem in projective space.

Van der Waerden, building on Poncelet’s ideas, published a rigorous modern version in 1928.

Weil (1946) and Serre (1957) subsequently published more general and precise results about intersections.

Potted History

Let P and Q be algebraic curves of degree dp and dq in a projective plane over an algebraically closed field. Then they intersect at precisely dpdq points, accounting for multiplicity.

We’ll proceed by unpacking each bit of jargon and seeing why it’s there.

As we do, the red words will gradually turn green.

Statement of Bézout’s Theorem

An algebraic curve can be defined using expressions called polynomials.

For our purposes, a polynomial is an expression that only involves numbers, positive whole number powers of x and y, and the operations of addition,, subtraction and multiplication.

Examples:

PolynomialsWe will not be considering curves that are described by “transcendental” functions like sin, cos, tan, log and ex. In the C18 Taylor showed using calculus that these can be approximated by polynomials, but to get equality (in the limit) you need the polynomial to have “infinite degree”.

Zeroes of Polynomials

A point (x, y) on the plane is a zero of a polynomial if when you plug the x and y coordinates into the polynomial, the resulting value is zero.

The zero points of a single polynomial usually – but not always – fall in curved lines.

An algebraic curve is a curve that can be described as the set of zeroes of a polynomial.

So we’ve decoded our first bit of jargon: Let P and Q be algebraic curves of

degree dp and dq in a projective plane over an algebraically closed field. Then they intersect at precisely dpdq points, accounting for multiplicity.

From now on we’ll identify a curve with its polynomial.

Algebraic Curves

The degree of a polynomial is the highest power is contains. If there are “cross terms” like x5y4, the degree is the sum of the powers (in this case, 9).

So we’ve decoded our second bit of jargon: Let P and Q be algebraic curves of

degree dp and dq in a projective plane over an algebraically closed field. Then they intersect at precisely dpdq points, accounting for multiplicity.

Degree of a Polynomial

Here are two algebraic curves. The ellipse is the one we saw

before, y2 + 2x2 — 7. The straight line is x — y.

Bézout's Theorem is about the points where they intersect. At these points both polynomials are zero.

The ellipse has degree 2 and the line has degree 1. Bézout's Theorem predicts they intersect at 2x1=2 points, which they do! Yay!

Intersections of Curves

Things aren’t looking so good here. All we’ve done is shifted the ellipse

and line relative to each other; this does not change their degrees. So Bézout's Theorem still predicts two intersections.

In the top image, there seem to be no intersections.

The bottom image seems to have only one.

So is Bézout's Theorem false?

All Is Not Well

Worse than this, some perfectly innocent-looking polynomials don’t have any zeroes at all.

For example:x2 + y2 – 1 is a circle, butx2 + y2 + 1 is an “empty curve” with no points!

If we have to worry about things like this all the time, our results are going to be full of messy caveats and special cases. That will make them harder to understand and to do things with.

Even Worse…

This problem will be “solved” by the three pieces of jargon we haven’t yet looked at: Let P and Q be algebraic curves

of degree dp and dq in a projective plane over an algebraically closed field. Then they intersect at precisely dpdq points, accounting for multiplicity.

The Solution…

A field is algebraically closed if every non-constant polynomial has at least one zero.

The field of real numbers, which we’ve been working with, is not algebraically closed.

Algebraic Closure

Above are the curves y—x2+1 and y+2. Where they intersect both must be zero at the same time.

The zeroes of y—x2+1 are points (√(y+1), y) for any choice of y.

The zeroes of y + 2 are points (x, -2) for any choice of x. So the second equation forces us to choose y = -2. But then

the first equation requires us to have x = √-1. BUT in the real number system, there are no square roots of

negative numbers!

Solving this problem involves our first “cheat”. We simply invent a new number and call it i. We

define i= √-1. Now our equations on the previous slide have a

solution: the points (i, —2) and (—i, —2). The curve y—x2+1 has degree 2, and y—x has degree 1,

so Bézout's Theorem predicts 2 points of intersection. And we have them! Yay!

It’s not quite as simple as that, because we need to make sure we’re still working over a field.

That means we need not just i but all the arithmetic combinations of real numbers with i.

This is called formed the field extension of by i. The result is called the field of complex numbers, denoted .

Complex Numbers

Is it legitimate to just invent a new number? Indeed, we’ve conjured a whole number system out of nowhere!

How can we possibly say these two curves intersect at two points? Where are they?

This is not obvious, but the complex numbers are two-dimensional. So a curve involving two sets of complex numbers lives in four-dimensional space.

We can no longer make pictures of these curves; in a sense they’ve become “ideal” or “abstract” objects.

Wait a Minute…

Like it or not, we’ve at least cleared up one more bit of jargon: Let P and Q be algebraic curves

of degree dp and dq in a projective plane over an algebraically closed field. Then they intersect at precisely dpdq points, accounting for multiplicity.

But as you can see, we have more problems in store… and we’ll solve them with more cheating (or abstract mathematics, depending on how you look at it).

Here are two algebraic curves: x + y – 4 and x + y + 4

They have degree 1, so Bézout's Theorem says they should have one point of intersection.

Where do they intersect?x + y – 4 = 0 when x + y = 4x + y + 4 = 0 when x + y = -

4These are both true only when 4 = – 4 .

Even the complex numbers can’t help us with that!

A Problem with Parallels

To get us out of this pickle, we repeat the trick that worked last time: we simply invent the things we need.

In this case, we need a point where the parallel lines cross. We call it a “point at infinity”.

NB two straight lines only meet at one point at infinity; it doesn’t matter “which way you go”.

We invent one point at infinity for each pair of parallel lines. Taken together, we call them the line at infinity.

The projective plane is the ordinary Euclidean plane plus the line at infinity. From now on, all our curves live in this projective space.

Since our coordinates are complex numbers, we call this the complex projective plane and write .

Of course, there’s more to it than that; but that’s all the detail we need for now.

Points at Infinity

Let P and Q be algebraic curves of degree dp and dq in a projective plane over an algebraically closed field. Then they intersect at precisely dpdq points, accounting for multiplicity.

Here are two algebraic curves of degree 2.

Bézout's Theorem says they should intersect at 2x2 = 4 points.

We can see one of them here, but instead of crossing over the curves are “just touching”.

Or can we? Is it an intersection? Is there more than one intersection point there? They seem to “cross but end up

on the same side they started on” – does that mean they “cross twice at a single point”?

“Kissing” Curves

For contrast, here are two very similar curves – the blue one has just been moved up a bit.

Here we see two nice, clear intersection points.

At the point on the right I’ve added, very faintly, a tangent line to each curve.

Intuitively, the tangent line is the straight line that points “in the same direction as the curve” at that point.

If you want to know more about tangents, sign up for our course on calculus after Christmas!

Note that the curves are going in different directions at this point, “cutting across” each other.

This is called a transversal intersection.

Transversality

Going back to the “kissing” curves, we see something different – the tangent lines of the two curves are the same.

This is called a non-transversal intersection. It’s a very “fragile” situation: any tiny perturbation will cause the curves to separate.

Guillemin & Pollack: “The naive point-set condition of intersection is seldom stable, and therefore it is meaningless in the physical world. Transversality, a notion that at first appears unintuitively formal, is all we can really experience.”Non-Transversal

Intersection

We define the intersection number of two curves at a point P to be:

0 if they don’t meet at P (duh); 1 if they intersect transversally at P; Calculated by black magic otherwise.

Intuitively, imagine using very small “tweaks” to the curves to get them to intersect transversally, then count the intersections. But that’s very rough!

If you really want grisly details on this, see here: https://etd.ohiolink.edu/!etd.send_file?accession=oberlin1385137385&disposition=inline

Counting each intersection not just as “one” but by its intersection number is what we mean by “accounting for multiplicity”.

We don’t really think the “kissing” curves cross each other “twice at the same point”, but this number does keep track of the fact that something decidedly different is happening here.

Multiplicity

Let P and Q be algebraic curves of degree dp and dq in a projective plane over an algebraically closed field. Then they intersect at precisely dpdq points, accounting for multiplicity.

Now this all makes sense! And it’s true!

But does it describe the geometrical situation we first started with? Or have we taken leave of reality in the process?

Karl Von Staudt was a key figure in the development of projective geometry over the complex numbers. His first book was published in 1847.

He believed that geometry should always be done synthetically – starting from qualitative, intuitively clear definitions and with little or no use of algebra, measurement or coordinates.

His “descriptive” style of argument relies on carefully-defined terms and logical deduction rather than visual images. It is uncompromising and very difficult to follow.

This contrasted with the very analytical approach of his contemporaries (e.g. Möbius and Plücker).

His work influenced Felix Klein and Sophus Lie, two of the founders of modern geometry.

Its rigorously logical style, emphasis on transformations and its qualitative approach (similar to topology) “look” very modern.

Yet it eschewed the power of the algebra, which would be decisive for later geometers.

His work also influenced Frege and Russell.

Karl von Staudt

As Coolidge puts it, von Staudt worried about the following problems in the geometry of his predecessor, Jakob Steiner:

The lingering presence of measurements and ratios of lengths, which he believed should be eliminated completely; and

The use of complex numbers. “Imaginary numbers in arithmetic can be defined as real number pairs, but what can we say of imaginary points except that they are not real?” (Coolidge, p.98).

Von Staudt’s strenuous efforts to resolve these problems indicate that the expansion of the ideas of “points” and “space” in geometry didn’t happen in an unprincipled way.

His engagement with the problem of imaginary numbers is largely in his later work of 1856-60.

What he really wanted to do was separate the geometry from the algebraic system of numerical coordinates – his intuition was that there was something purely geometrical that the imaginary numbers were describing, and that could be accessed directly, as it were, by purely synthetic means.

Although he did manage to give a construction of imaginary points using so-called “elliptic involutions” of lines, it wasn’t entirely satisfactory. (Wilson has the most comprehensible explanation of this we’ve found).

History has, so far, fallen largely on the side of considering algebra and geometry to be complementary rather than opposed.

Karl von Staudt

Von Staudt’s treatment of “points at infinity” is rather different.

Consider that “being a mother” is really a relationship between two people: “X is the mother of Y”. From this, though, we can derive the abstract noun “motherhood”. We haven’t created any new objects by doing this, but rather added a “concept-object” to our framework for understanding the world.

Similarly, “being parallel” is a relationship between two lines. But from this we can extract the concept-object that corresponds to their shared direction; this, von Staudt suggests, is what we mean by a “point at infinity”. There is no real “infinity” being appealed to here at all.

Von Staudt always sought to drive out blind calculation with rigorous, conceptual arguments. He did not trust algebra (which was mechanical) or visual intuition (which was flawed). He was one of the early proponents of a logical approach to mathematics.

Karl von Staudt

We’ve been silently assuming that our two curves don’t have any “common components” – meaning roughly there aren’t whole chunks of them that overlap exactly.

For example, a circle has degree 2, so Bézout's Theorem predicts that two circles will intersect at 2x2=4 points in total. But if the two circles are actually two copies of the same circle, in exactly the same place, they intersect at an infinite number of points!

(And their tangent lines agree at every point, so those are non-transversal intersections.)

For our purposes this is a “pathological case” and we ignore it. We’re mentioning it here just for the sake of completeness.

Technical Caveat

Abhyankar, S. S. (2001) “What is the difference between a parabola and a hyperbola?” in Wilson & Gray (ed.) Mathematical Conversations: Selections from the Mathematical Intelligencer. London: Springer.

Coolidge, J.L. (1940) A History of Geometrical Methods. Oxford: Clarendon Press.

Dieudonné, J. (1974) History of Algebraic Geometry. Monterey, CA: Wadsworth. (NB A less technical essay-length version can be found online.)

Guillemin & Pollack (1974) Differential Topology. New Jersey: Prentice Hall.

Reich, K. (2005) “Karl Georg Christian von Staudt, Book on Projective Geometry (1847)” in Grattan-Guiness (ed.), Landmark Writings in Western Mathematics, 1640-1940. Amsterdam: Elsevier.

Rowe, D.E. (2001) “Imaginary elements in 19th-century geometry” in Lutzen (ed.) Around Caspar Wessel and the Geometric Representation of the Complex Numbers. Denmark: Det Kongelige Danske Videnskabernes Selskab.

Wilson, M. (2010) ”Frege’s mathematical setting” in Potter & Ricketts (ed.) The Cambridge Companion to Frege. Cambridge: Cambridge University Press.

Bibliography