[Paper Review] Metric properties of semialgebraic mappings

Krzysztof Kurdyka, Stanisław Spodzieja+1 more|arXiv (Cornell University)|Dec 16, 2014

Advanced Differential Equations and Dynamical Systems17 references4 citations

TL;DR

This paper establishes effective upper and lower bounds for Łojasiewicz exponents in the context of semialgebraic sets and mappings, proving that the local and global Łojasiewicz exponents of an overdetermined semialgebraic mapping $ F: X \to \mathbb{R}^m $ on a closed semialgebraic set $ X \subset \mathbb{R}^n $ (with $ m > \dim X $) coincide with those of the composition $ L \circ F: X \to \mathbb{R}^k $, where $ k = \dim X $ and $ L $ is a generic linear map. The results generalize complex-analytic estimates to the real semialgebraic setting with explicit degree-based bounds.

ABSTRACT

We give an effective estimation from above for the local Łojasiewicz exponent for separation of semialgebraic sets and for a semialgebraic mapping on a closed semialgebraic set. We also give an effective estimation from below of the Łojasiewicz exponent in the global separation for semialgebraic sets and estimation of the Łojasiewicz exponent at infinity of a semialgebraic mapping similar to the Jelonek result in the complex case.

Motivation & Objective

• To extend global and local Łojasiewicz exponent estimates from complex algebraic geometry to the real semialgebraic setting.
• To provide effective, degree-based upper and lower bounds for Łojasiewicz exponents in separation problems involving semialgebraic sets.
• To establish that the Łojasiewicz exponent of an overdetermined semialgebraic mapping $ F: X \to \mathbb{R}^m $ equals that of $ L \circ F: X \to \mathbb{R}^k $ for generic linear $ L: \mathbb{R}^m \to \mathbb{R}^k $, where $ k = \dim X $.
• To generalize classical results such as those of Jelonek, Cygan, and Kollár to the real semialgebraic case with effective quantitative control.

Proposed method

• Derive effective upper bounds for the local Łojasiewicz exponent of separation of semialgebraic sets and for semialgebraic mappings on closed semialgebraic sets using degree-based analysis of defining polynomials.
• Establish effective lower bounds for the global Łojasiewicz exponent in semialgebraic separation and at infinity for semialgebraic mappings, analogous to Jelonek's complex results.
• Use generic linear projections $ L: \mathbb{R}^m \to \mathbb{R}^k $ with $ k = \dim X $ to reduce overdetermined mappings $ F: X \to \mathbb{R}^m $ to minimal-dimensional mappings $ L \circ F: X \to \mathbb{R}^k $.
• Leverage Zariski openness and density of generic linear maps to ensure that the zero sets and asymptotic behavior of $ F $ and $ L \circ F $ agree locally and at infinity.
• Apply results from semialgebraic geometry, including properness of mappings on semialgebraic sets and properties of generic linear projections, to relate the Łojasiewicz exponents of $ F $ and $ L \circ F $.
• Use inequalities involving distance to zero sets and norms of mappings to compare $ |F(x)| $ and $ |L(F(x))| $, establishing equivalence of Łojasiewicz exponents via comparison constants.

Experimental results

Research questions

• RQ1 What effective upper bound can be given for the local Łojasiewicz exponent of separation of two semialgebraic sets or of a semialgebraic mapping on a closed semialgebraic set?
• RQ2 What effective lower bound can be established for the global Łojasiewicz exponent in the separation of semialgebraic sets or for the Łojasiewicz exponent at infinity of a semialgebraic mapping?
• RQ3 How does the Łojasiewicz exponent of an overdetermined semialgebraic mapping $ F: X \to \mathbb{R}^m $ relate to that of its composition $ L \circ F: X \to \mathbb{R}^k $ with a generic linear map $ L $?
• RQ4 Can the Łojasiewicz exponent at infinity of a semialgebraic mapping be bounded from below in terms of degrees of defining polynomials, generalizing complex results to the real case?

Key findings

• An effective upper bound for the local Łojasiewicz exponent of separation of semialgebraic sets and of a semialgebraic mapping on a closed semialgebraic set is established in terms of the degrees of the defining polynomials.
• An effective lower bound for the global Łojasiewicz exponent in the separation of semialgebraic sets is derived, analogous to Jelonek's complex result but in the real semialgebraic setting.
• The Łojasiewicz exponent at infinity of a semialgebraic mapping $ F: X \to \mathbb{R}^m $ is bounded from below in terms of the degrees of the polynomials defining $ F $ and $ X $, generalizing Cygan's and Jelonek's complex estimates.
• For an overdetermined semialgebraic mapping $ F: X \to \mathbb{R}^m $ with $ m > \dim X $, the local and global Łojasiewicz exponents of $ F $ are equal to those of $ L \circ F $ for a generic linear map $ L: \mathbb{R}^m \to \mathbb{R}^k $, where $ k = \dim X $.
• The Łojasiewicz exponent at infinity of $ F $ is preserved under generic linear projection $ L $, provided $ F^{-1}(0) $ is compact, and the exponent is bounded below by a degree-dependent expression.
• The equivalence of Łojasiewicz exponents between $ F $ and $ L \circ F $ is proven via comparison of norms and zero sets using genericity and properness arguments in semialgebraic geometry.