Some Degree-Eight M-Curves Exceed Patchworking’s Reach

Combinatorial patchworking is one of the main constructive tools for producing real algebraic curves with prescribed topology. The question in this paper is whether that tool is universal for plane real schemes in low degree: given an ambient isotopy type of a real plane curve, can it always be realized as a T-curve from a triangulated Newton polygon and a sign distribution? The answer is almost yes in small degree, and then sharply no at degree eight.

The paper resolves a question of Itenberg and Viro by proving that every nonempty real scheme of degree $d \leq 7$ can be obtained by combinatorial patchworking, but some maximal degree-eight schemes cannot. The obstruction is not a failure of existence of the algebraic curves themselves. It is a failure of the patchworking construction to reach all of them.

Core Contribution

The main negative result is a classification constraint for maximal T-curves of degree eight. For an $M$-curve of degree eight, the maximal number of real components is $M=22$, and the paper tracks the pair $(p,n)$ of even and odd ovals. The authors prove that a maximal degree-eight T-curve can have only one of four pairs:

• $(19,3)$
• $(15,7)$
• $(11,11)$
• $(7,15)$

This immediately excludes known degree-eight $M$-curves with $(p,n)=(3,19)$ from the T-curve class. Viro and Shustin had constructed such curves; the paper’s Corollary 22 shows that none of them can be produced by combinatorial patchworking. That is the central separation: real algebraic realizability is strictly larger than T-realizability at degree eight.

Technical Approach

The proof works by reducing the topology of maximal T-curves to controlled changes induced by Harnack splits. The authors use Haas’s theorem to start from a special Harnack curve and then analyze how collections of splits alter the oval parity counts. A key intermediate step is Lemma 16, which shows that any zone decomposition of a collection of splits is valid for some patchwork. This lets the argument focus on the combinatorics of zones and surgical twists rather than on a fixed triangulation.

For degree eight, Theorem 17 permits the authors to ignore even splits when counting even and odd ovals. Lemma 18 then narrows the relevant odd simple or double Harnack splits to seven equivalence classes. Each such split has a computable effect on $(p,n)$: most contribute $m=4$, while one simple split, corresponding to $(x,y)=(7,5)$ in the proof, contributes $m=8$. The paper then uses nesting and disjointness of zones to bound the total possible effect by $12$, yielding the four permitted pairs in Theorem 21.

This is a clean mathematical obstruction. It does not depend on the Gudkov–Rohlin congruence; the authors explicitly point out that the proof of Theorem 21 is purely combinatorial.

Positive Classification Through Degree Seven

The other half of the paper is constructive. For degrees below eight, the authors give explicit patchworking data showing that every nonempty real scheme occurs as a T-curve. The degree-seven section relies on Viro’s classification, where degree seven has $M=16$ and 121 possible real schemes. Theorem 38 states that all 121 schemes can be generated as T-curves from four regular triangulations of $7\cdot\Delta_2$, named cen, spl, fra, and hon, by varying sign distributions.

The paper reports the individual coverage of those four triangulations: cen realizes 115 types, spl realizes 107, fra realizes 103, and hon realizes 47. The degree-seven proof data records, for each real scheme, the numbers $p$, $n$, $p+n$, the triangulation used, and the sign distribution $\sigma$. Earlier tables give analogous sign-distribution certificates for lower degrees.

Results and Analysis

The strongest result is the combination of an exhaustive positive construction up to degree seven and a proof obstruction at degree eight. This matters because patchworking is often treated as a highly flexible generator of real schemes. The paper identifies the first degree where that flexibility provably ends for plane curves.

The degree-eight result is especially precise: it does not merely exhibit one failed construction. It proves that the entire class of maximal degree-eight T-curves is confined to four oval-parity pairs. Since algebraic $M$-curves with $(3,19)$ already exist, the conclusion follows without relying on computation-heavy search over patchworks.

The constructive side is also substantial, but its evidence is more certificate-like than conceptual: the paper supplies triangulations and sign strings that realize all listed schemes. That is appropriate for a classification theorem, though it means the reader must trust or verify the enumeration machinery and the correspondence between sign distributions and real schemes. The limitation is not in the theorem statements; it is in portability. The approach proves a low-degree boundary, but it does not by itself give a general characterization of which higher-degree real schemes fail to be T-curves.