Analytic preparation, cells and curves

Four mathematical readings with editable sources and native MathML:

• Analytic finiteness and preparation: convergent division and finiteness, analytic units and coordinate changes, the full two-coordinate reduction and dimension induction, global analytic cells, complement and convergent Puiseux expansions with parameters.
• Weierstrass preparation and division: preparation, division with constants independent of the dividend, Noetherianity and four solved exercises.
• Curve selection and Łojasiewicz inequalities: value-level gradient bounds, residual inequalities, two-sided analytic curves, a separate radial estimate and definable choice, with six complete exercise solutions.
• Exercises on preparation: why a bounded function can require an inverse power, and how even substitutions interact with parameter-dependent poles. Both have complete solutions.

The preparation proof specifies the classical projective product definition of global subanalytic sets. Its dimension induction is closed; parameterized series retain analytic coefficients, positive variable radii and the even cleared exponents needed for a two-sided analytic substitution. The curve reading uses these proved inputs.

The analytic finiteness reading is original CC0 text citing Guillaume Valette’s On subanalytic geometry, arXiv:2507.23622v1, for its theorems and proof routes. The curve-selection reading was independently reconstructed by GPT-6 Astra (OpenAI), Ultra, with new exposition and six solved exercises dedicated to CC0. It credits Valette’s classical results and the common selection, Puiseux and chain-rule methods; its mathematical and source-structure review is specific to that reading. The preparation exercises are original CC0 text citing Valette for the Exercise 15 example; Exercise 16 includes the October 2026 correction distinguishing the original and ramified variables. The Weierstrass preparation and division reading is original CC0 text; Jean-Pierre Demailly’s Complex Analytic and Differential Geometry, 21 June 2012, is further reading for it. Jean-Marie Lion and Jean-Philippe Rolin are cited for the projective product definition of global subanalytic sets.

These readings are a selection from Constructible and perverse sheaves. Self-checked by the writing AI.

Attribution and reuse terms · Provenance · Preparation source · Division source · Curve source · Exercise source

Rebuild with Python 3 and Pandoc: run python build/build_reader.py in this directory. Wide formulas scroll within the page on small screens.

The analytic finiteness reading also proves that cell dimension is intrinsic, that definable maps do not raise dimension, that closure commutes with fibres after a finite partition of the base, that the frontier has smaller dimension than the set, and that fibrewise continuous functions are continuous on finitely many base pieces. Three exercises with complete solutions accompany these results. The frontier is distinguished from the topological boundary. Analytic regularity, uniformization and resolution are not treated in these readings.