A loop that cannot miss the middle

Some polynomials have no roots. That is the first thing anyone learns about them: x² + 1 is never zero, and the graph sits above the axis with a comfortable margin. It stops being true the moment the imaginary unit is admitted, and then it stops being true so completely that the result has a name reserved for the results that close a subject — every polynomial of degree n has exactly n roots, counted properly, and there is nothing left to look for.

The usual proofs of that are analytic and none of them is short. The argument here is a picture.

The argument is in the difference between the first panel and the last, and it takes three observations to close.

What the picture is doing #

Take a circle of radius r centred at the origin. Every point on it is a complex number; feed each one to the polynomial; the outputs trace a closed curve, because the inputs came back to where they started. That curve can be anywhere and any shape, but one question about it always has an integer answer: how many times does it go round the origin?

That integer is the winding number, and it is obtained by walking the curve and adding up the angle it turns through as seen from the origin, then dividing by a whole turn. The figure computes it that way — fourteen hundred and forty samples, each contributing the small angle since the last — and asserts two things that would invalidate the count if they failed: that no sample lands exactly on the origin, where the angle is undefined, and that no single step turns more than a quarter of a turn, which would mean the sampling was too coarse to know which way the curve went round.

The winding number cannot be fractional. The curve is closed, so the accumulated angle is a whole number of turns, and the figure checks that too.

Small circles #

Feed a very small circle to a polynomial and almost nothing happens. Every term but the constant one is tiny, so the output is a small wobble around the constant term — around p(0). If that constant is not zero, the wobble stays well away from the origin, and a loop that never comes near the origin does not go round it. The winding number is 0.

The first panel of the figure is that case: the circle of radius four tenths, whose image is a small closed curve sitting near 2 and enclosing nothing.

The condition is exactly the one stated: p(0) must not be zero. If it is, the search is over before it starts — zero is a root, and there is nothing to prove.

Large circles #

Far from the origin, one term of a polynomial swamps the rest. On a circle of radius 10, the cubic term of z³ − 2z + 2 has size a thousand, while the other two terms together cannot exceed twenty-two. The image is therefore a slightly perturbed copy of the image of z³ — and the image of z³ on a circle is a circle traversed three times.

That last step is the whole mechanism, and it is the first rung of the complex-number ladder: multiplying adds angles. A point that goes once round the origin has its argument increase by a full turn; its square’s argument increases by two turns; its cube’s, by three. The winding number of the image of a large circle under a polynomial of degree n is n, and the perturbation from the lower terms cannot change it, because a perturbation that never reaches as far as the origin cannot change how many times a curve goes round it.

The roots of unity are the cleanest instance. A number whose fifth power is 1 must have an angle that five copies of add to a whole number of turns, so the roots are equally spaced on the unit circle, and the polynomial z⁵ − 1 has its five roots visible without any algebra at all.

The dominance argument deserves a number rather than a gesture. On a circle of radius r, the leading term of a degree-three polynomial has size r³ and the rest has size at most the sum of the other coefficients times r², so the leading term wins as soon as r exceeds that sum. For z³ − 2z + 2 the sum is four, so every circle of radius above four is safely in the regime where the cubic term dictates the shape. The figure uses radius 2, which is already enough — the bound is generous rather than tight, and being generous is what makes it easy to state.

The step that cannot be smooth #

Two facts are now in hand: the winding number is 0 for small circles and n for large ones. The remaining question is what happens in between, and the answer is that it must jump.

The winding number is an integer. As the radius grows continuously, the image curve moves continuously. An integer that changes has to jump, and the only way a winding number can jump is for the curve to pass through the origin — because as long as the curve stays clear of the origin, it can be deformed continuously without the count changing, and a continuous deformation of an integer that is not allowed to jump does not change it at all.

So at some radius between the small one and the large one, the image curve passes through the origin. A point of the curve at the origin is a point z with p(z) = 0.

That is the theorem.

The figure above says something stronger than the theorem needs, and the strength is checked rather than claimed: at every radius, the winding number equals the number of roots inside that radius. The roots are found by an entirely separate method — an iteration that converges to all of them at once — and the two numbers are compared at each circle. They agree.

What the count is measuring, exactly #

It is worth pausing on what kind of object the winding number is, because it is the first quantity on this page that belongs to the shape rather than to the numbers.

Two curves that can be deformed into one another without ever touching the origin have the same winding number. That is not a theorem about polynomials; it is the definition of the invariant being useful. It means the number survives any amount of wobbling, stretching or reparametrising, and it changes only when the curve is dragged across the point it is counting around.

Invariants of that kind are what topology is for, and this site has met several. The Euler characteristic survives any subdivision of a surface. Tricolourability survives the three legal moves on a knot diagram. Each is a quantity computed from a particular drawing that turns out not to depend on the drawing, and each is used the same way: two things with different values are certainly different, and the value cannot be argued with.

The winding number’s particular usefulness here is that it is an integer that varies continuously with the radius — which is a contradiction unless it does not vary at all, and that contradiction is the entire proof.

A stronger statement, and an awkward one #

The winding count is not merely nonzero somewhere. It counts. A circle enclosing two roots has an image that goes round twice; a circle enclosing all n goes round n times. That is why the theorem can promise exactly n roots rather than at least one: peel off a root, divide the polynomial by the corresponding linear factor, and repeat on a polynomial of one lower degree.

The awkwardness is in what the argument does not deliver. It proves a root is there and gives no way of finding it. The radius at which the count jumps is located by watching the count jump — which requires being able to evaluate the polynomial, which is fine — but the point on that circle where the curve crosses the origin is not produced by anything above. The theorem is an existence proof, and existence proofs are the ones that leave a reader holding nothing.

This is the same discomfort that runs through the pigeonhole principle, which proves a collision exists without saying which pair collides, and through Brouwer’s fixed point, which promises a point that does not move and points at none. It is a genuine feature of these arguments rather than a failure of exposition: the machinery that proves the existence is a count, and a count has no address in it.

The degree, and where it goes wrong #

The quartic z⁴ − 1 is the case that shows the count is not obliged to be gentle. Its four roots are all at distance exactly 1 from the origin, so as the radius grows past 1 the winding number goes from 0 to 4 in a single step.

That is also the case where the family refuses to draw. Asked for the image of the circle of radius exactly 1, the generator stops the build: the curve passes through the origin, the angle at that sample is undefined, and a winding number computed through it would be a number with no meaning. The refusal is the assertion doing its job, and the radius it refuses is precisely the interesting one.

There is one more edge worth naming. A polynomial of degree zero — a nonzero constant — has winding number 0 at every radius, and the argument correctly proves nothing, because there is nothing to prove. A constant polynomial has no roots and the theorem does not claim otherwise.

The same argument, one dimension down #

The structure of the proof is worth separating from the complex numbers, because it is not really about them.

A real continuous function that is negative somewhere and positive somewhere else must be zero in between. Nobody finds that surprising, and its proof is the same shape as the one above: a quantity that can only take certain values — here the sign, there the winding number — is one thing at one end and another at the other, and it cannot change without passing through the forbidden state.

That is the intermediate value theorem, and it is why every odd-degree real polynomial has a real root: far to the left it is large and negative, far to the right large and positive. Even-degree polynomials get no such promise, which is exactly why x² + 1 was allowed to be the counterexample this essay opened with.

What the complex plane adds is a second dimension for the curve to move in, and with it a richer invariant. A sign has two values and can only report a crossing; a winding number has infinitely many and reports how many crossings, which is what upgrades some root exists to exactly n roots exist. The dimension the argument gains is the dimension the answer gains.

What the picture cannot show #

The picture is of one polynomial. Four panels of a cubic and four of a quartic are not a proof about all polynomials, and nothing on this page is. What the figures show is the mechanism at a size where every step can be checked; the argument itself is three sentences long and quantifies over everything.

The continuity is assumed, not drawn. The claim that a winding number cannot change without the curve crossing the origin is where all the real work lives, and it is a topological statement about the plane with a point removed. It is the same statement that makes the index of a zero an invariant, and it cannot be got from a picture of four circles.

And the multiplicity is invisible. A polynomial with a double root has a curve that winds twice as the circle grows past it, and nothing in the drawing distinguishes that from two roots close together. The two are genuinely different situations — one has a repeated factor and the other does not — and the winding count is deliberately blind to the difference, which is why the theorem says counted with multiplicity and why the count is well behaved.

A note on what the roots were found with #

The figure that compares winding numbers against root counts needs the roots, and it gets them from an iteration that starts with a ring of guesses and refines them all together until each one satisfies the polynomial to within a millionth of a millionth. That iteration is checked: every value it returns is fed back into the polynomial and the result is required to be nearly zero.

But the iteration takes for granted the thing this essay is proving. It starts with as many guesses as the degree, on the assumption that there are that many roots to find, and it would run just as confidently on a polynomial with fewer. So the comparison in that figure is a corroboration and not a proof, and it is worth saying which way round the dependence goes: the winding argument establishes that the roots are there, and the iteration is then entitled to go looking for them.

Reversing that order is a real mistake and an easy one, because the iteration is the thing that produces numbers and numbers feel like evidence. What the iteration actually produces is three points at which the cubic is very nearly zero, which is not the same as a proof that it is exactly zero anywhere.

Where the ladder goes next #

The winding number is the first integer in the subject that is attached to a shape rather than to a number, and once noticed it is everywhere. It is the index of a zero of a vector field, which is what makes combing a sphere flat impossible. It is the degree of a map from a circle to a circle. It is what separates the two sides of a curve in the plane, which is the subject of inside and outside and a genuinely different question from the one it looks like.

The rung directly above this one is about what the theorem costs. Knowing that n roots exist says nothing about writing them down, and for degree five and above there is provably no formula in radicals — a result that is about symmetry rather than about existence, and one that needs the group of a polynomial rather than a picture of a loop. The two results sit oddly together and are both true: every quintic has five roots, and no expression built from its coefficients with arithmetic and radicals can name them.