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, Cauchy division, Noetherianity and a solved multiplicity exercise.
• Curve selection and Łojasiewicz inequalities: definable choice, analytic curve selection, arc and compact inequalities, radial estimates and the gradient inequality.
• 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.

Guillaume Valette’s On subanalytic geometry, arXiv:2507.23622v1, is adapted under CC BY 4.0. Jean-Pierre Demailly’s Complex Analytic and Differential Geometry, 21 June 2012, supplies the division reading under his custom OpenContent grant. Human credit, reuse terms and identified AI additions remain visible. Lion–Rolin’s primary definition is cited as research context; their article is not imported or relicensed.

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 preparation reading now also proves intrinsic cell dimension, dimension nonincrease under definable maps, closure after base partitioning, strict frontier decrease and parameter continuity, with three further complete solutions. The frontier is distinguished from the topological boundary. This new block has a bounded independent mathematical review; no independent review of the whole collection is claimed. Analytic regularity, uniformization and resolution remain separate work in the parent course.