Bézout’s Theorem
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Philosophy of Mathematics
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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:
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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
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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
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Motivation
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.
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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
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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.
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We’ll proceed by unpacking each bit of
jargon and seeing why it’s there.
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As we do, the red words will gradually
turn green.
Statement of Bézout’s
Theorem
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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:
𝑥 2 + 2𝑦 − 7
𝑦 2 + 2𝑥 2 − 7
𝑥3𝑦2 + 𝑥2 − 𝑥
Polynomials
We 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”.
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.
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Zeroes of Polynomials
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:
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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
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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).
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So we’ve decoded our second bit of
jargon:
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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
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Here are two algebraic curves.
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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
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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 innocentlooking polynomials don’t have any
zeroes at all.
 For example:
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x2 + y2 – 1
x2 + y2 + 1
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is a circle, but
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…
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This problem will be “solved” by
the three pieces of jargon we
haven’t yet looked at:
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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…
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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.
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!
Algebraic Closure
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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).
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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.
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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 ℝ by i. The result is
called the field of complex numbers, denoted ℂ.
Complex Numbers
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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?
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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…
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Like it or not, we’ve at least cleared up one
more bit of jargon:
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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).
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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 = 4
x + y + 4 = 0 when x + y = -4
These are both true only when 4 = – 4 .
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Even the complex numbers can’t
help us with that!
A Problem with Parallels
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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”.
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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.
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Since our coordinates are complex numbers, we call this the
complex projective plane and write 𝑃2 ℂ.
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Of course, there’s more to it than that; but that’s all the detail we need for now.
Points at Infinity
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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?
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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
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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.
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Intuitively, the tangent line is the straight line
that points “in the same direction as the
curve” at that point.
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If you want to know more about tangents, sign up for our
course on calculus after Christmas!
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Note that the curves are going in different
directions at this point, “cutting across” each
other.
This is called a transversal intersection.
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Transversality
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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
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We define the intersection number of two curves at a point P
to be:
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0 if they don’t meet at P (duh);
1 if they intersect transversally at P;
Calculated by black magic otherwise.
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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
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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?
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Karl Von Staudt was a key figure in the development of
projective geometry over the complex numbers. His first
book was published in 1847.
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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 carefullydefined 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.
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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
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As Coolidge puts it, von Staudt worried about the following problems in
the geometry of his predecessor, Jakob Steiner:
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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.
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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).
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History has, so far, fallen largely on the side of considering algebra and
geometry to be complementary rather than opposed.
Karl von Staudt
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Von Staudt’s treatment of “points at infinity” is rather different.
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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
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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!
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(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
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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