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Constructing Discontinuous but Locally Bounded Rational Functions using \L ojasiewicz Inequalities

This paper demonstrates that for real multivariate polynomials $P$ and $Q$ vanishing at a point where the zero set of $Q$ is contained in that of $P$ , there exists a locally bounded rational function of the form $P^p/Q^q$ whose continuous extension to the zero set of $Q$ is necessarily discontinuous, a result established using Lojasiewicz inequalities.

Original authors: Adam Coffman, Yifei Pan

Published 2026-06-02

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Adam Coffman, Yifei Pan

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to cross a river. Usually, if you have a bridge (a mathematical function), you can walk across it smoothly. But sometimes, the bridge has a hole right in the middle. If you try to walk over the hole, you might fall, or the path might become so steep you can't climb it.

In the world of mathematics, specifically with rational functions (which are just fractions made of polynomials, like $\frac{x^2}{y}$ ), mathematicians often look for "holes" where the bottom number (the denominator) becomes zero. Usually, if the top number (numerator) is also zero at that spot, we might be able to patch the hole and walk across smoothly.

This paper, by Adam Coffman and Yifei Pan, is about finding a very specific, tricky kind of hole. They are looking for bridges that are safe to walk on (bounded) but impossible to cross smoothly (discontinuous).

Here is the breakdown of their discovery using simple analogies:

1. The "Safe but Bumpy" Bridge

Imagine a bridge that is perfectly flat and safe to stand on, but if you try to walk toward a specific point in the middle, the ground suddenly shifts. You can get infinitely close to the center, and you will never fall off the edge (the function is bounded), but you can never step onto the center point without the ground jumping up or down (the function is discontinuous).

The Classic Example: Think of the function $f(x,y) = \frac{xy}{x^2+y^2}$ . If you walk toward the center (0,0) in a straight line, you might arrive at a height of 0.5. But if you walk in a different straight line, you arrive at 0. If you walk in a curve, you might arrive at 1. The bridge is "safe" (it never goes to infinity), but it's a mess because the height depends entirely on your direction.

2. The Big Question

The authors ask: Can we always build a bridge like this?
They start with two shapes (polynomials), let's call them $P$ (the numerator) and $Q$ (the denominator). Both shapes touch the ground (equal zero) at a specific point. They know that the "hole" in $Q$ is completely inside the "hole" in $P$ .

They want to know: Is there always a way to tweak the powers of these shapes so that the resulting fraction is safe (bounded) but still bumpy (discontinuous)?

3. The Magic Tool: The "Lojasiewicz Inequality"

To answer this, the authors use a powerful mathematical tool called the Lojasiewicz Inequality.

Think of this inequality as a speed limit sign for how fast one shape can shrink compared to another.

Imagine $P$ and $Q$ are two balloons deflating as you approach the center.
The inequality tells us that $P$ cannot deflate too much faster than $Q$ without breaking the rules of geometry.
There is a specific "ratio" or "exponent" (like a gear setting) where $P$ shrinks just enough to keep the fraction from exploding (bounded), but not quite enough to smooth out the bumps (discontinuous).

4. The Main Discovery

The paper proves that yes, this tricky bridge always exists.

No matter what two shapes ( $P$ and $Q$ ) you start with (as long as they both hit zero and $Q$ 's hole is inside $P$ 's), you can always find a specific "gear ratio" (a rational number) to mix them.

If you mix them with this ratio, the function stays within safe limits (it won't shoot to infinity).
However, if you try to extend the function to the center point, it will still be jagged and discontinuous.

It's like finding the perfect amount of sugar to add to coffee: enough to keep it from being too bitter (bounded), but not enough to make it taste like water (continuous).

5. How They Build These Bridges

The authors show two main ways to construct these "safe but bumpy" functions:

Method A: The Gradient Trick (The Slope Detector)
They take a shape $P$ and look at its "gradient" (which is like measuring the steepness of a hill at every point). They create a fraction where the top is the hill's height ( $P$ ) and the bottom is the steepness squared ( $|\nabla P|^2$ ).
- Analogy: Imagine a mountain. The top is the height, the bottom is how steep the slope is. Near the peak, the height and steepness are related in a specific way. By adjusting the "gear ratio" (the exponents), they create a function that stays bounded but refuses to be smooth at the very peak.
Method B: The Complex Number Trick (The Shadow)
They use complex numbers (numbers with an "imaginary" part) to create real-world functions.
- Analogy: Imagine shining a light on a 3D object to cast a 2D shadow. They take a complex function, split it into its "real" and "imaginary" parts, and use those as the numerator and denominator. This creates a function that spins around the center point. As you get closer to the center, the value spins wildly, never settling on a single height, even though it never flies off to infinity.

6. Why This Matters (According to the Paper)

The paper doesn't claim this will fix bridges in the real world or cure diseases. Instead, it solves a puzzle in Real Algebraic Geometry.

It confirms that the behavior of these mathematical functions is more complex than we thought. Even when a function is "well-behaved" enough to stay within a box (bounded), it can still be "ill-behaved" enough to refuse to connect smoothly at a point.

In summary: The authors proved that you can always engineer a mathematical fraction that is safe from infinity but refuses to be smooth, using a specific "speed limit" rule (Lojasiewicz inequality) to find the perfect recipe.

Technical Summary: Constructing Discontinuous but Locally Bounded Rational Functions

Problem Statement
The paper addresses the behavior of real multivariate rational functions $f = P/Q$ near points where both the numerator $P$ and denominator $Q$ vanish (the indeterminacy set). Specifically, the authors investigate the relationship between local boundedness and continuity. While it is well-known that rational functions can be bounded yet discontinuous (e.g., along different paths approaching the origin), the paper seeks a general constructive method to demonstrate that such functions always exist under specific geometric conditions. The core question is: Given polynomials $P$ and $Q$ vanishing at a point $\vec{0}$ such that the zero set of $Q$ is contained in the zero set of $P$ ( $Z(Q) \subseteq Z(P)$ ), does there always exist a rational power of the function, specifically of the form $P^{2p}/Q^{2e}$ , that is locally bounded but whose extension (defined as 0 on $Z(Q)$ ) is discontinuous?

Methodology
The authors employ tools from real algebraic geometry and the theory of semi-algebraic sets. The methodology relies heavily on:

Lojasiewicz Inequalities: The proofs utilize the classical Lojasiewicz inequality and the Lojasiewicz gradient inequality. These inequalities provide bounds on the relationship between the values of polynomials and their gradients (or other polynomials) near their zero sets. Specifically, the existence of a minimal rational exponent $\beta_0$ (or $\theta_0$ ) such that $|P|^\beta \le C|Q|$ is central to the argument.
Semi-algebraic Geometry: The authors analyze the structure of the image of polynomial maps and the geometry of zero sets. They use properties of semi-algebraic sets, including the local description of real algebraic curves (Proposition 3.4) and the dimension of indeterminacy sets.
Constructive Exponent Selection: The paper constructs specific rational exponents $p/e$ derived from the minimal Lojasiewicz exponent. The strategy involves showing that for exponents slightly larger than the minimal bound, the function becomes unbounded, while for the minimal bound, the function is bounded but fails to be continuous due to differing limits along specific paths (often derived from Puiseux expansions).
Complex Constructions: In Section 6, an alternative construction is presented using complex polynomials. By taking the real or imaginary parts of ratios of holomorphic polynomials (e.g., $f(z)^\nu / \overline{g(z)}^\mu$ ), the authors generate bounded real rational functions with discontinuities without relying on the specific real algebraic geometry machinery used in the main theorems.

Key Contributions and Results
The paper establishes several theorems and lemmas that characterize the existence and construction of these functions:

Main Existence Theorem (Theorem 5.2): If $P$ and $Q$ are polynomials vanishing at $\vec{0}$ with $Z(Q) \subseteq Z(P)$ , there exist positive integers $p$ and $e$ such that the rational function $P^{2p}/Q^{2e}$ is locally bounded near $\vec{0}$ , but its extension to $Z(Q)$ (setting the value to 0) is discontinuous. The ratio $p/e$ is unique and corresponds to the minimal Lojasiewicz exponent relating $P$ and $Q$ .
Gradient-Based Construction (Theorem 7.4): A specific and robust method is provided where the denominator is chosen as a power of the squared norm of the gradient of the numerator, i.e., $Q = |\nabla P|^{2q}$ . Theorem 7.4 proves that if $p/q$ equals the minimal Lojasiewicz exponent $\theta_0$ for $P$ at a critical point, the resulting function is bounded but discontinuous.
Dimensionality of Indeterminacy Sets (Theorem 7.10): The authors demonstrate that this construction is not limited to isolated zeros. They show that for any real algebraic set $A$ strictly contained in a hyperplane, one can construct a bounded rational function whose indeterminacy set is exactly $A$ . This contrasts with the generic case where bounded rational functions often have indeterminacy sets of codimension 2 or higher; here, the indeterminacy set can have dimension $n-2$ or larger.
Complex Variable Constructions (Section 6): The paper provides a method to generate bounded, discontinuous rational functions in $\mathbb{R}^{2m}$ by taking the real or imaginary parts of expressions like $h(z)/\overline{h(z)}$ , where $h$ is a holomorphic polynomial. This yields functions with indeterminacy sets of dimension $2m-2$ .

Significance and Claims
The paper claims to provide a general, elementary (in the sense of using foundational real algebraic geometry) proof that the phenomenon of "bounded but discontinuous" is not an anomaly but a guaranteed property for a wide class of rational functions defined by polynomial pairs with nested zero sets.

The authors emphasize that their approach clarifies the precise threshold for continuity. They show that the existence of a bounded, discontinuous rational function of the form $P^{2p}/Q^{2e}$ is equivalent to a strong version of the Lojasiewicz inequality for the pair $(P, Q)$ . The work connects the analytic property of continuity with the algebraic property of the minimal Lojasiewicz exponent.

The paper does not propose new applications in computer algebra or engineering but notes that the problem of computing limits of multivariable rational functions remains of interest in computer algebra. The significance lies in the theoretical construction: providing a systematic way to generate counterexamples to the intuition that local boundedness implies continuity (or that continuity can be restored by simple extension) in multivariable calculus. The results refine the understanding of the "gap" between boundedness and continuity, showing that for any such pair of polynomials, there is a specific rational power that sits exactly on the boundary of continuity.