Two closed sets in the plane that don't intersect have positive distance. But how does the distance behave as you approach the points where the sets nearly touch? The Łojasiewicz separation exponent quantifies this: for semi-analytic sets A and B with distance function d(x, A∪B), the distance separating A from B decreases as a power law near the point of closest approach, with exponent α.
The exponent is a local invariant of the pair (A, B) at the point of near-contact. Computing it requires understanding the geometry of both sets and their relative positioning — not just their individual shapes but how they approach each other.
The paper computes the separation exponent for real semi-analytic sets in two dimensions. The computation reduces to analyzing the Puiseux expansions of the sets' boundaries near the point of near-contact. Each boundary is a curve expressible as a fractional power series, and the separation exponent is determined by the leading-order disagreement between the two series.
The result gives the exponent as a rational number computable from the Newton polygons of the defining functions. The rationality is a consequence of the semi-analytic structure — the boundaries have algebraic complexity, and the exponent inherits this algebraicity. In the real-analytic case, the exponent is always a ratio of natural numbers derivable from the curve's parametrization.
The separation exponent measures the difficulty of distinguishing nearby sets — a large exponent means the distance vanishes slowly (sets are easily separated), and a small exponent means the distance vanishes quickly (sets are hard to separate). The invariant is geometric, computable, and rational.