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Sums of squares on curves and surfaces

This paper investigates sums of higher even powers in the coordinate rings of singular planar curves defined by $x^M=y^m$ , demonstrating that unlike the quadratic case, the cone of sums of squares can have codimension 2.

Original authors: Bartłomiej Bychawski, Bartosz Głowacki, Tomasz Kowalczyk

Published 2026-06-23

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CC BY 4.0

Original authors: Bartłomiej Bychawski, Bartosz Głowacki, Tomasz Kowalczyk

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 have a giant, infinite toolbox filled with numbers and shapes. In the world of mathematics, specifically Real Algebraic Geometry, there's a game called "The Sum of Squares."

Here's the basic rule: Can you build a specific number (or a function) by adding up a bunch of other numbers raised to a power? For example, can you make the number 10 by adding up squares (like $1^2 + 3^2$ )?

The paper you're asking about is like a team of mathematicians (Bychawski, Głowacki, and Kowalczyk) exploring the limits of this game. They are asking: "Is there a limit to how many pieces we need to build any shape?"

Here is a breakdown of their findings using simple analogies:

1. The "Pythagoras Number" (The Limit of Pieces)

Think of the Pythagoras number as the maximum number of Lego bricks you might ever need to build a specific structure.

If the number is finite, it means no matter how complex the shape, you can always build it with, say, no more than 100 bricks.
If the number is infinite, it means there are some shapes so complex that you might need an endless supply of bricks to build them, or perhaps no fixed limit exists at all.

2. The Curves: When Things Get "Bumpy"

The authors first looked at singular curves. Imagine a smooth road (a normal curve) versus a road with a sharp, jagged spike (a singular curve).

The Finding: On these jagged, spike-filled roads, as the spikes get more extreme, the number of "bricks" (powers) needed to build certain shapes grows without bound.
The Metaphor: It's like trying to build a house on a mountain of jagged rocks. The more jagged the rocks get, the more bricks you need to level the ground. They proved that for these specific jagged curves, the number of bricks needed can become infinitely large.

3. The Surfaces: Adding "Square Roots"

Next, they looked at surfaces (like sheets of paper floating in space) that are defined by adding "square roots" of polynomials. Think of this as adding a new, strange dimension to your toolbox.

The Finding: If you have a collection of these strange surfaces that all touch at a single point (a common zero) and behave nicely in a specific direction, the number of bricks needed to build shapes on them is infinite.
The Metaphor: Imagine a group of mirrors that all meet at a single point. If you try to reflect a complex image using these mirrors, you might find that no matter how many mirrors you stack, you can't perfectly recreate the image without an infinite number of them.

4. The "Regulous" Functions: The Smooth Patch

This is the most surprising part of the paper. They studied a special type of function called 0-regulous functions.

The Analogy: Imagine a piece of fabric that is mostly smooth, but has a few tiny, invisible tears. A "regular" function is a perfect, unbroken fabric. A "0-regulous" function is a fabric that is continuous (you can't feel the tear) but might be mathematically "rational" (made of fractions) in a way that allows it to be patched perfectly.
The Finding: On these special, patchable fabrics, the number of bricks needed is finite.
Why it matters: This is a big deal because, until now, the only places where mathematicians knew the brick count was finite were very simple places (like fields or valuation rings). They found a new family of complex mathematical spaces where the limit is actually manageable. It's like discovering a new type of terrain where, despite the complexity, you never need more than a specific number of bricks to build anything.

5. The "Bad Set": Where the Math Breaks Down

Finally, they looked at the "Bad Set."

The Concept: Sometimes, to build a shape using fractions (rational functions), you have to divide by something. If that "something" becomes zero, the math breaks. The "Bad Set" is the collection of points where these denominators vanish.
The Finding: For simple squares (the quadratic case), these "bad spots" are usually very small (like a single point on a line). But for higher powers (like 4th powers or 6th powers), the "bad spots" can be much larger—they can form a whole line or a surface (codimension 2).
The Metaphor: If you are building with simple blocks, the only place you might trip is a single pebble. But if you are building with complex, high-power blocks, the "pebbles" can turn into a whole pothole in the road.

Summary

The paper is a map of mathematical territory. It tells us:

Jagged curves and certain surfaces are so complex that the number of pieces needed to build things there is infinite.
However, on special "patchable" surfaces (0-regulous), the number of pieces is finite, which is a rare and valuable discovery.
When dealing with higher powers, the "danger zones" (bad sets) where math breaks down can be much larger and more complex than we previously thought.

They didn't find a way to build a bridge to the real world (like in engineering or medicine); instead, they mapped out the rules of the game itself, showing us exactly where the rules get infinite and where they stay finite.

Technical Summary: Sums of Squares on Curves and Surfaces

Problem Statement
The paper investigates the finiteness of higher Pythagoras numbers for specific rings arising in real algebraic geometry. For a commutative ring $R$ with identity and an integer $n > 0$ , the $2n$ -Pythagoras number, denoted $p_{2n}(R)$ , is defined as the smallest integer $g$ such that any element expressible as a sum of $2n$ -th powers can be written as a sum of at most $g$ such powers. If no such $g$ exists, $p_{2n}(R) = \infty$ . While the 2-Pythagoras number (sums of squares) is well-studied, with known finiteness for dimension 1 affine $\mathbb{R}$ -algebras and infiniteness for excellent rings of dimension $\ge 3$ , the behavior of higher even powers ( $2n$ for $n > 1$ ) is less understood. The authors focus on three specific contexts: coordinate rings of singular planar curves, coordinate rings of algebraic surfaces defined by adjoining square roots of polynomials, and rings of 0-regulous functions.

Methodology
The authors employ a combination of algebraic techniques, degree analysis, and constructive counterexamples:

Degree Bounds and Projections: For singular curves defined by $x^M = y^m$ , the authors utilize the isomorphism between the coordinate ring and $R[t^m, t^M]$ . They construct a linear map $\psi$ to lift representations from the singular curve back to the polynomial ring $R[x, y]$ , preserving degree bounds to establish lower bounds on Pythagoras numbers.
Automorphism Groups and Character Theory: For rings of the form $R[x, y, \sqrt{f_1}, \dots, \sqrt{f_n}]$ , the authors analyze the group of automorphisms (sign changes of the square roots). By applying character orthogonality relations, they isolate specific components of polynomial representations to derive contradictions regarding the existence of finite sums of powers.
Weighted Degree Functions: In proving infiniteness for higher powers on surfaces, a specialized degree function $\deg_A$ is defined on the extended ring to track the growth of terms under specific substitutions (e.g., $y = x^{r_L}$ ), allowing for a descent argument.
Regulous Function Theory: For 0-regulous functions, the proof relies on the extension properties of continuous rational functions. The authors utilize the fact that strictly positive 0-regulous functions on open constructible sets are units and can be extended to the whole variety under specific continuity conditions.
Explicit Construction: To analyze the "bad set" (the common zero set of denominators in representations of a polynomial as a sum of $2n$ -th powers), the authors construct specific families of polynomials and analyze their behavior in formal power series rings.

Key Contributions and Results

Singular Planar Curves: The paper generalizes a known result regarding unbounded Pythagoras numbers for rings of the form $R[t^a, t^{2a-1}]$ . The authors prove that for a sequence of coprime pairs $(m_i, M_i)$ with $m_i < M_i$ and $m_i \to \infty$ , the $2n$ -Pythagoras numbers of the coordinate rings $R[X_{m_i, M_i}]$ (where $X_{m,M} = \{x^M = y^m\}$ ) tend to infinity as $i \to \infty$ . This holds for any $n \ge 1$ , though the case $n=1$ was previously known to be finite for fixed $m, M$ .
Surfaces and Adjoined Square Roots: The authors study rings $R[x, y, \sqrt{f_1}, \dots, \sqrt{f_n}]$ corresponding to surfaces in $\mathbb{R}^N$ . They prove that if $f_1, \dots, f_n$ are square-independent, have a common zero, and possess positive $y$ -leading coefficients, then the 2-Pythagoras number of the ring is infinite. This result is extended to higher even powers ( $2s$ for $s > 1$ ) under similar conditions, showing $p_{2s}(R[x, y, \sqrt{f_1}, \dots, \sqrt{f_n}]) = \infty$ .
0-Regulous Functions: The paper establishes a significant finiteness result for the ring of 0-regulous functions, $R_0(X)$ , on an irreducible 0-regulous variety $X$ . The authors prove that $p_{2n}(R_0(X)) < \infty$ for all $n$ . This provides the first known family of rings (excluding fields and valuation rings) where all higher even Pythagoras numbers are finite. The proof relies on the ability to extend sums of powers from open sets to the whole variety.
Bad Sets of Higher Order: The authors investigate the codimension of the "bad set" $Bad_{2n}(f)$ , defined as the zero set of all possible denominators when representing a polynomial $f$ as a sum of $2n$ -th powers of rational functions. While it is known that for $n=1$ (sums of squares) on nonsingular surfaces, the codimension is at least 3, the authors demonstrate that for $n > 1$ , the codimension can be exactly 2. They provide explicit examples of polynomials where the bad set is a single point (codimension 2 in $\mathbb{R}^2$ ).

Significance and Claims
The paper claims to advance the understanding of higher Pythagoras numbers by:

Identifying specific geometric conditions (singular curves, surfaces with common zeros) that force these invariants to be infinite, contrasting with the finite nature of the 2-Pythagoras number in dimension 1.
Providing the first example of a non-trivial class of rings (0-regulous functions) where all higher even Pythagoras numbers are finite, supporting the conjecture that finiteness for one even power implies finiteness for all.
Correcting the intuition derived from the quadratic case regarding the size of the bad set, showing that for higher orders, the set of "bad points" can be larger (codimension 2) than in the quadratic case (codimension $\ge 3$ ).

The authors conclude by posing an open problem regarding the relationship between $p_2(R)$ and $p_{2n}(R)$ , specifically asking if $p_2(R) \le p_{2n}(R)$ holds generally, noting that while known examples follow this rule, no general proof exists.