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A note on the real Jacobian conjecture in degree 7

This paper establishes the validity of the real Jacobian conjecture for degree 7 polynomials whose highest degree homogeneous part takes the form $\alpha x^7 + \beta x^6y$ , while also proving the non-existence of atypical Jacobian pairs where the degree of $p$ is 7 and the degree of $q$ is even and coprime to 7.

Original authors: Tomasz Kowalczyk

Published 2026-08-13

📖 3 min read🧠 Deep dive

CC BY 4.0

Original authors: 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 are a cartographer trying to draw a map of a magical, infinite landscape. In this world, every location is defined by two numbers, like a street address and a floor number. Now, imagine you have a special machine—a "polynomial map"—that takes any point in this landscape and shuffles it to a new spot. The big question mathematicians have been asking for decades is: If this machine never squashes two different points into the same spot (making the map "injective"), does it guarantee that the machine is safe to use everywhere? Specifically, if the machine's internal gears (its "Jacobian") never jam or stop turning, can we be 100% sure that no two starting points ever end up in the same destination?

This puzzle is known as the Real Jacobian Conjecture. It's like checking if a complex, twisting slide in a playground is safe. If the slide never gets stuck (the gears never stop), does that mean a kid sliding down will never get stuck on top of another kid? For a long time, mathematicians knew the answer was "yes" for simple, small slides. But for bigger, more complicated slides, they discovered that sometimes, even if the gears don't jam, the slide can still get stuck in a weird way, trapping two kids in the same spot. These are called "atypical" slides. The big mystery has been: How complicated does a slide have to get before it can do this? Is there a specific size where the magic stops working?

This paper by Tomasz Kowalczyk dives into that exact question, focusing on slides made of polynomials with a specific "degree" of 7. Think of "degree" as the complexity or the number of twists in the slide. The author investigates a specific type of slide where the most complex part (the highest-degree terms) looks exactly like $\alpha x^7 + \beta x^6 y$ (where $\alpha$ and $\beta$ are numbers that aren't both zero). By using a clever geometric tool called a "Newton polygon" (which is basically a shape drawn by connecting the dots of the slide's most important parts), the author proves that for these specific degree-7 slides with this particular highest-degree structure, the Real Jacobian Conjecture holds true. In other words, if the slide is this specific shape and the gears don't jam, it is impossible for two points to get stuck together.

Furthermore, the paper tackles a second, related riddle: What if the second part of the machine (the "q" part) has an even number of twists and doesn't share any common factors with the number 7? The author proves that in this scenario, too, the machine is safe. There are no "atypical" pairs where the first part is degree 7 and the second part is an even number coprime to 7. This helps narrow down the search for the smallest possible "broken" slide. While we know a broken slide exists with a degree of 7 and another part of degree 29, this paper shows that you won't find a broken one if the second part is an even number like 17, 19, or 21. The author has effectively ruled out a whole bunch of possibilities, bringing us one step closer to finding the absolute smallest, simplest counterexample to this mathematical mystery.

Problem Statement
The paper addresses the Real Jacobian Conjecture, which posits that a polynomial map $(p, q): \mathbb{R}^2 \to \mathbb{R}^2$ with a non-vanishing Jacobian determinant (a Jacobian pair) must be injective. While it is known that atypical Jacobian pairs (counterexamples to injectivity) exist, the specific degree conditions under which they may occur remain a subject of investigation. The author focuses on the case where the degree of the first polynomial, $p$ , is 7. Previous work established that if $\deg(p) \leq 6$ , the pair is always typical (injective). Recent results have shown the existence of an atypical pair with $\deg(p)=7$ and $\deg(q)=29$ . This paper investigates whether atypical pairs exist for $\deg(p)=7$ under specific constraints on the highest degree homogeneous part of $p$ and the parity of $\deg(q)$ .

Methodology
The author employs a geometric approach centered on the Newton polygon of the polynomial $p$ . Key methodological tools include:

Newton Polygon Analysis: The study of the convex hull of the support of $p$ ( $\Delta_p$ ), specifically focusing on "outer edges" and their associated "outer normal vectors."
Branches at Infinity: Utilizing the parametrization of branches of $p(x,y)=0$ at infinity to relate them to the edges of the Newton polygon.
Degeneracy and Submersion Properties: Analyzing whether $p$ is "degenerate" on specific edges (possessing multiple factors coprime with $xy$) and applying lemmas regarding polynomial submersions.
Case-by-Case Enumeration: Systematically listing and eliminating possible Newton polygon configurations for polynomials of degree 7 with specific leading terms.
Coordinate Transformations: Applying affine changes of variables (e.g., $x := x + a$ or $y := y + cx^2$ ) to reduce complex cases to previously solved lower-degree scenarios or standard forms.
Sign Change Arguments: Using Lemma 1.1 to demonstrate that if the sum of certain vertices of the Newton polygons of $p$ and $q$ has an even coordinate, the Jacobian changes sign, thereby ruling out the existence of a Jacobian pair.

Key Contributions and Results
The paper establishes the following main results:

Typicality for Specific Leading Terms (Degree 7):
The author proves that if $(p, q)$ is a Jacobian pair with $\deg(p)=7$ , the pair is typical (injective) if the highest degree homogeneous part ( $p_+$ ) takes one of the following forms:
- $p_+ = \alpha x^7$ (where $\alpha \neq 0$ ).
- $p_+ = \beta x^6 y$ (where $\beta \neq 0$ ).
- $p_+ = \alpha x^7 + \beta x^6 y$ (where $\alpha^2 + \beta^2 \neq 0$ ).
The proof involves an exhaustive analysis of all possible Newton polygons for these cases. The author demonstrates that for every possible configuration, either the pair is typical, or the existence of a Jacobian mate $q$ is impossible (e.g., by reducing the problem to a degree 6 case, which is known to be typical, or by showing the Jacobian must change sign).
Non-Existence of Atypical Pairs with Even Degree $q$ :
The paper proves that if $(p, q)$ is a Jacobian pair with $\deg(p)=7$ and $\deg(q)$ is even and coprime to 7, then the pair is typical.
- The proof utilizes a linear change of variables to assume $x$ divides the leading term of $p$ .
- By analyzing the vertices of the Newton polygons of $p$ and $q$ , the author shows that the sum of these vertices results in a vector with exactly one even coordinate.
- According to Lemma 1.1, this implies the Jacobian changes sign, contradicting the assumption that $(p, q)$ is a Jacobian pair.

Significance and Scope
The paper contributes to the resolution of the degree conditions for the Real Jacobian Conjecture. By combining ideas from previous works (specifically [4] and [12]), the author narrows the search space for potential counterexamples.

The results confirm that no atypical Jacobian pairs exist for $\deg(p)=7$ if the leading term is of the form $\alpha x^7 + \beta x^6 y$ or if $\deg(q)$ is even and coprime to 7.
The paper explicitly references the known counterexample with $\deg(p)=7$ and $\deg(q)=29$ (from [4]) and the computational check showing no counterexample exists for $\deg(q) \leq 15$ (from [2]).
The author concludes that to find the smallest possible degree of an atypical pair with $\deg(p)=7$ , future work must focus on determining the existence of atypical mates for the specific polynomial $p_7 = x + y + 3xy + 3x^2y + 2x^2y^2 + 3x^3y^2 + x^4y^3$ with degrees in the set $\{17, 19, 21, 23, 25, 27, 28\}$ .

The paper does not claim to fully resolve the conjecture for degree 7 but provides significant constraints that eliminate large classes of potential counterexamples, thereby refining the open problem.

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