We’ve seen that semialgebraic Teddy and semialgebraic Lambkin are both images of a four-dimensional ball. Could they be images of each other?
The Teddy-Lambkin Theorem
Ursula Whitcher
Mathematical Reviews (AMS)
My given name means bear—it’s the same root as the constellation Ursa Major—so growing up I had a huge collection of teddy bears. In this month’s Feature Column, I want to tell you about a theorem that uses teddy bears to provide a new perspective on a fundamental question: how weird can polynomials be?
A typical precalculus sequence treats lines as the simplest possible functions, followed by polynomials of degree two and higher. Based on this early education, most of us assume that polynomials are not very complicated. But the polynomials we know best are of very low degree and depend on only one variable. Higher degree, multivariable polynomials can exhibit startling and counterintuitive behavior. Some of the most famous open problems in modern mathematics, including the P vs NP problem in computer science and the Hodge conjecture, ask for specific measurements of how weird polynomials and their solutions can become.
Our subject today is real algebraic geometry. In other words, we’re interested in studying polynomials whose coefficients are real numbers and their solutions in $\mathbb{R}^n$. These are the kinds of polynomials you met in your first algebra and precalculus courses. But even the simplest real polynomials force us to deal with alternative scenarios and special cases that don’t come up over the complex numbers. For example, we can always use the quadratic formula to find two (possibly identical) complex solutions to $x^2 + bx + c =0$. But if we want real solutions, this might be impossible!
What does it mean to consider a region cut out by polynomials? One option is to consider only the solutions to polynomial equations. For example, the unit circle is given by solutions to $x^2+y^2-1=0$ in the plane. For many questions, properties of the unit disk, including the interior of the circle, are just as important. To deal with situations like these, real algebraic geometers often study semialgebraic sets. These are finite unions and intersections of regions defined by finite systems of equations of the form $P(x_1,\dots, x_n) = 0$ or $P(x_1, \dots, x_n) > 0$, where $P$ is a polynomial. In other words, we allow both polynomial equations and polynomial inequalities. In this framework, the unit disk is given by the union of solutions to $x^2+y^2-1=0$ and $-x^2-y^2+1>0$.
An annulus—the region between two circles—is another simple example of a semialgebraic set. We can cut out a convex polygon using the linear equations for its edges, or cut out a polyhedron using the planes that form its faces. If we’re feeling artistic, we can also make more complicated shapes.
Some semialgebraic shapes in the plane
One advantage of semialgebraic sets over the solutions to real polynomial equations is that semialgebraic sets play nicely with familiar functions. For example, consider the hyperbola in $\mathbb{R}^2$ described by $xy – 1 = 0$. If we take the image of the hyperbola under the projection map $\pi: \mathbb{R}^2 \to \mathbb{R}$ given by $\pi(x,y) = x$, we get all of the real line except for 0. We cannot write this set as a finite union of solutions to real polynomial equations, because every real polynomial in one variable has a finite number of solutions, and our set has an infinite number of points. However, because we can write it as the union of solutions to $x>0$ and $-x>0$, the projection of the hyperbola is a semialgebraic set.
The image of the hyperbola under projection to the $x$-axis.
The Tarski-Seidenberg theorem, named for the twentieth-century mathematicians Alfred Tarski and Abraham Seidenberg, states that the same pattern holds in any dimension: the projection $\pi: \mathbb{R}^{n+1} \to \mathbb{R}^n$ of a semialgebraic set is always a semialgebraic set. (Like many mathematicians, Tarski was an immigrant: he left his home country of Poland just before the German and Soviet invasion in 1939, settled in the USA, and did not see his wife and children again until World War II was over.)
A circle of Spanish mathematicians including José F. Fernando, José Manuel Gamboa, and Carlos Ueno, has been studying the images of semialgebraic sets in greater depth. Their work offers strategies for turning the question about weird polynomials we started with into precise mathematical statements.
In 2023, Fernando and Ueno used “bricks” including spheres and half-spheres, ellipsoids, cylinders, and tetrahedra to construct semialgebraic sets in $\mathbb{R}^3$ that could be realized as polynomial images of the unit ball in $\mathbb{R}^4$. One of them is a semialgebraic teddy bear and another is a semialgebraic sheep. I’ll illustrate with my own models made from polymer clay.
Clay models of a semialgebraic bear and sheep.
Fernando and Ueno referred to the bear and sheep by the German diminutives “Bärchen” and “Schäfchen” (as commemorated, for example, in a German schoolteacher’s YouTube channel). I like to use the English “Teddy” and “Lambkin”. Mathematically, these shapes have some special properties. They are compact subsets of $\mathbb{R}^3$. In other words, they are closed (they contain all of their boundary points) and bounded (they fit within a sphere of finite radius). Fernando and Ueno also impose the technical condition that their bricks be connected along analytic paths.
We’ve seen that semialgebraic Teddy and semialgebraic Lambkin are both images of a four-dimensional ball. Could they be images of each other? Antonio Carbone investigated this problem as part of a PhD project at the Università di Trento that Fernando supervised. In 2024, Carbone and Fernando published a paper in the Advances of Mathematics that shows the answer is yes—if we’re willing to use the right kind of function.
The functions in question are Nash maps, named for the multifaceted mathematician and winner of the Nobel prize in economics John Forbes Nash Jr. Let’s define them in two stages.
Suppose $f: \mathbb{R}^m \to \mathbb{R}^n$ is a function. We say $f$ is a semialgebraic map if its graph $\{(x,y) \in \mathbb{R}^{m+n} \mid y = f(x)\}$ is a semialgebraic set. We can use the Tarski-Seidenberg theorem to conclude that the images of semialgebraic maps are semialgebraic sets by projecting onto the final $n$ coordinates.
A semialgebraic map is called a Nash map if it is also a smooth map. We’re using “smooth” here in the sense of multivariable calculus, where we require that the matrix of partial derivatives at each point has full rank. (In particular, when the source and target dimensions are the same, the matrix of partial derivatives at each point is square, and we simply need it to be invertible.) Intuitively, smooth maps can’t introduce sharp creases or excessively pointy bits.
We are now ready to state Carbone and Fernando’s Bärchen-Schäfchen Theorem—or, as I like to think of it, the Teddy-Lambkin Theorem.
Teddy-Lambkin Theorem.
Let $\mathscr{S} \subset \mathbb{R}^m$ be a semialgebraic set of dimension $d$ and let $\mathscr{T} \subset \mathbb{R}^n$ be a compact semialgebraic set connected by analytic paths of dimension $e$. Suppose $e \leq d$. Then there exists a Nash map $f: \mathbb{R}^m \to \mathbb{R}^n$ such that $f(\mathscr{S}) = \mathscr{T}$.
In the case of our semialgebraic teddy bear and semialgebraic sheep, we have $d=e=3$, so there is a Nash map from the bear to the sheep and another from the sheep to the bear!
The maps in question don’t have to be one-to-one, so more intense transformations are possible. We could use a Nash map to convert a semialgebraic coffee cup shape into a teddy bear or make a lambkin out of a filled (jelly?) donut. The realm of real polynomials is—no jokes about it—very, very strange.
Further Reading
- Antonio Carbone and José F. Fernando. Surjective Nash maps between semialgebraic sets. Advances in Mathematics, Volume 438, February 2024, 109288.
- José F. Fernando and Carlos Ueno. On polynomial images of a closed ball. J. Math. Soc. Japan 75(2): 679-733 (April, 2023). DOI: 10.2969/jmsj/88468846