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Real critical points of $T$-polynomials that are sums of squared monomials and topology of $T$-hypersurfaces

Aloïs Demory

TL;DR

The paper investigates real critical points of $T$-polynomials that are sums of squared monomials and the topology of the associated $T$-hypersurfaces built by combinatorial patchworking. By linking real critical points to lattice-point data and complex Hodge numbers via patchworking and Morse theory, it derives asymptotic upper bounds on sums of Betti numbers and analyzes the sharpness of these bounds. It also shows that, in general, $T^2$-hypersurfaces do not admit asymptotically maximal families, while contrasting SQM-hypersurfaces can achieve stronger growth in certain Betti numbers. The work provides explicit constructions that achieve or approach these bounds, including detailed triangulations and Newton-polytopes, and demonstrates sharpness for extrema and the number of connected components, with extensions to higher dimensions and first Betti numbers. Together, these results clarify the limits of real topology achievable by $T$- and SQM-patchworked hypersurfaces and illuminate the role of combinatorial data in governing real loci.

Abstract

We study the topology of the real algebraic hypersurfaces in $\mathbb{P}^n$ that can be constructed via combinatorial patchworking using triangulations that are dilations by two of other triangulations. By examining the real critical points of the polynomials that define such hypersurfaces, we find some asymptotical upper bounds on various sums of their Betti numbers. We then discuss the sharpness of those bounds.

Real critical points of $T$-polynomials that are sums of squared monomials and topology of $T$-hypersurfaces

TL;DR

The paper investigates real critical points of -polynomials that are sums of squared monomials and the topology of the associated -hypersurfaces built by combinatorial patchworking. By linking real critical points to lattice-point data and complex Hodge numbers via patchworking and Morse theory, it derives asymptotic upper bounds on sums of Betti numbers and analyzes the sharpness of these bounds. It also shows that, in general, -hypersurfaces do not admit asymptotically maximal families, while contrasting SQM-hypersurfaces can achieve stronger growth in certain Betti numbers. The work provides explicit constructions that achieve or approach these bounds, including detailed triangulations and Newton-polytopes, and demonstrates sharpness for extrema and the number of connected components, with extensions to higher dimensions and first Betti numbers. Together, these results clarify the limits of real topology achievable by - and SQM-patchworked hypersurfaces and illuminate the role of combinatorial data in governing real loci.

Abstract

We study the topology of the real algebraic hypersurfaces in that can be constructed via combinatorial patchworking using triangulations that are dilations by two of other triangulations. By examining the real critical points of the polynomials that define such hypersurfaces, we find some asymptotical upper bounds on various sums of their Betti numbers. We then discuss the sharpness of those bounds.
Paper Structure (12 sections, 17 theorems, 10 equations, 1 figure)

This paper contains 12 sections, 17 theorems, 10 equations, 1 figure.

Table of Contents

  1. Bounds on the Betti numbers of $T^2$-hypersurfaces
  2. Bounds on some sums of Betti numbers
  3. Non-existence of asymptotically maximal families
  4. A bound when $V(\tau(P))$ is bounded from above
  5. A bound when $V(\tau(P))$ is bounded from below
  6. Proof of non-existence of asymptotically maximal families of hypersurfaces
  7. Comparing $SQM$-hypersurfaces to $T^2$-hypersurfaces
  8. Sharpness of the bounds
  9. Number of extrema
  10. A better bound in the case of curves
  11. Number of connected components
  12. First Betti number

Key Result

Theorem 4.1

For any positive integer $k$, one has Additionally, one has $b_1(\mathbb{R} A) \stackrel{\mathclap{\normalfont\scriptsize\hbox{n}}}{\leq} h^{1,n-2}(\mathbb{C} A)$. Moreover, if $n \leq 6$, then for any non-negative integer $k\leq n$, one has

Figures (1)

  • Figure 1: The described triangulation and sign distribution for $\frac{m}{2}=4$. The color of the vertices represent their sign.

Theorems & Definitions (41)

  • Theorem 4.1
  • proof
  • Proposition 4.1
  • proof
  • Corollary 4.1
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • ...and 31 more