On the separation \L ojasiewicz exponents of real analytic sets in the real plane

1. Problem Statement

The paper addresses the computation and estimation of the separation Łojasiewicz exponent, denoted as $L(A, B)$, for two real analytic set germs $A$ and $B$ in $\mathbb{R}^2$ near the origin.

Given two real analytic functions $f$ and $g$ defining sets $V_f = f^{-1}(0)$ and $V_g = g^{-1}(0)$, the separation Łojasiewicz inequality states that there exist constants $C, r > 0$ and an exponent $\beta \geq 1$ such that:
$$d(x, V_f) + d(x, V_g) \geq C \cdot d(x, V_f \cap V_g)^\beta$$
for all $\|x\| \leq r$, where $d(x, X)$ is the Euclidean distance from point $x$ to set $X$. The separation Łojasiewicz exponent $L(V_f, V_g)$ is defined as the infimum of all such valid exponents $\beta$.

The specific challenges addressed are:

1. Computability: How to explicitly compute $L(V_f, V_g)$ for general real analytic functions using their defining properties.
2. Effectiveness: How to derive an explicit upper bound for $L(V_f, V_g)$ when $f$ and $g$ are polynomials, expressed in terms of their degrees.

2. Methodology

The authors employ a combination of Newton polygon techniques, Newton–Puiseux expansions, and a sliding algorithm adapted from Kuo and Parusinski.

Key Technical Tools:

• Newton Polygon Relative to an Arc: For a function germ $f$ and a Puiseux series $\phi(y)$, the authors define a Newton polygon $P(f, \phi)$ based on the expansion of $f(X + \phi(Y), Y)$. This allows the analysis of the function's behavior along specific curves.
• The Sliding Algorithm: An iterative process that starts with an initial Puiseux series $\phi_0$ and generates a sequence $\phi_0 \to \phi_1 \to \dots \to \phi_\infty$. This sequence converges to a root of the function (or a "final result" if no root exists). The algorithm adjusts the series based on the edges of the Newton polygon to increase the order of vanishing.
• Approximations of Puiseux Series:
  • Real Approximation ($\gamma_R$): If a Puiseux root has non-real coefficients, the authors define a "real approximation" by replacing the first non-real coefficient with a generic real number.
  • Approximation of Two Series ($\phi_{1,2}$): Defined as the $\rho$-approximation of two distinct series $\phi_1, \phi_2$, where $\rho = \text{ord}(\phi_1 - \phi_2)$. This captures the "contact" between two branches.
• Curve Selection Lemma: The authors utilize the fact that the supremum of the local exponents $\ell(\phi)$ over all semi-algebraic analytic curves $\phi$ passing through the origin determines the global separation exponent $L(V_f, V_g)$.

3. Key Contributions and Results

A. Formula for Real Analytic Functions (Theorem 3.1)

The primary theoretical contribution is a formula to compute $L(V_f, V_g)$ using the Newton–Puiseux roots of the product $f \cdot g$.

• Definitions:

  • Let $h = \gcd(f, g)$.
  • Let $V_a(f \cdot g)$ be the set of all real approximations of pairs of Newton–Puiseux roots of $f \cdot g$.
  • Define $\ell(\phi) = \frac{\min(\text{ord}_d(\phi, V_f), \text{ord}_d(\phi, V_g))}{\text{ord}_d(\phi, V_h)}$, where $\text{ord}_d$ represents the order of distance to the zero set.
  • Define $L_+(V_f, V_g) = \max \{ \ell(\gamma) \mid \gamma \in V_a(f \cdot g) \}$.
  • Define $L_-(V_f, V_g)$ similarly by considering the reflection $y \to -y$.

• Main Formula:
  $$L(V_f, V_g) = \max \{ L_+(V_f, V_g), L_-(V_f, V_g) \}$$
  This reduces the problem of finding the separation exponent to analyzing the contact orders of the real approximations of the roots of $f$ and $g$.

B. Proof Strategy for Theorem 3.1

The proof relies on a contradiction argument:

1. Assume there exists a curve $\phi$ such that $\ell(\phi) > \max(L_+, L_-)$.
2. Using Lemma 3.2, they show that if $\ell(\phi)$ is strictly larger than the maximum of the approximations, the Newton angles of $f$ and $g$ relative to $\phi$ must be equal, and the associated edge polynomials must have a unique real root.
3. This allows the construction of a "sliding" sequence that converges to a common root of $f, g,$ and $h$.
4. This convergence implies $\ell(\phi) = 1$, contradicting the assumption that $\ell(\phi) > 1$ (since the maximum of approximations is at least 1). Thus, the supremum is attained within the set of approximations.

C. Effective Estimation for Polynomials (Theorem 4.1)

For the case where $f$ and $g$ are polynomials of degree at most $d$, the authors provide an explicit, effective upper bound for the exponent.

• Result:
  $$L(V_f, V_g) \leq \frac{(2d-1)^2 + 1}{2}$$
• Derivation:
  • They construct $F = f^2 + g^2$, noting that $\deg(F) \leq 2d$ and $V_F = V_f \cap V_g$.
  • They apply the classical effective Łojasiewicz inequality for a single polynomial $F$ (citing results from [23]), which gives an exponent bound of $(2d-1)^2 + 1$.
  • By relating $|F(x)|$ to $d(x, V_g)$ and $d(x, V_f \cap V_g)$, they derive the square root of the classical bound, resulting in the exponent $\frac{(2d-1)^2 + 1}{2}$.
• Significance: This bound is sharper than previous results (specifically [18, Corollary 8]) for the two-variable case, particularly when the intersection $V_f \cap V_g$ is not an isolated point.

4. Significance and Applications

• Computational Efficiency: The paper moves beyond existence proofs to provide an algorithmic formula for computing the separation exponent. By reducing the problem to the analysis of Newton–Puiseux roots and their real approximations, it offers a concrete path for calculation in singularity theory.
• Bridge between Analysis and Algebra: The work connects the analytic properties of real analytic germs (via Puiseux series) with algebraic bounds (via polynomial degrees).
• Optimization and Geometry: The separation Łojasiewicz exponent is crucial in:
  • Optimization: Determining convergence rates of algorithms and analyzing the geometry of feasible sets.
  • Topology: Computing topological invariants of real algebraic sets.
  • Singularity Theory: Understanding the behavior of functions near singularities and at infinity.
• Refinement of Bounds: The effective bound provided for polynomials improves upon existing literature, offering tighter constraints for problems involving real algebraic sets in $\mathbb{R}^2$.

Conclusion

Hoang and Nguyen successfully characterize the separation Łojasiewicz exponent for real analytic sets in the plane. They establish that the exponent is determined by the "worst-case" contact order between the real approximations of the roots of the defining functions. Furthermore, they translate this theoretical insight into a practical, degree-dependent bound for polynomial systems, enhancing the toolkit available for researchers in real algebraic geometry and optimization.