Hauser's Exercises 3 and 4 on the order: the strata of constant order #
Appendix B of [Hau03] asks for the strata of constant order of two subschemes. Exercise 3:
f = x³ + yᵏzᵐ on 𝔸³; the examples instantiate the loci {ord ≥ 2} and {ord = 3} computed
in Hironaka/Scheme/IdealSheaf/Order/AffineSpace.lean at k = m = 3 (the top locus is the union of
the two axes {x = y = 0} ∪ {x = z = 0}) and at k = m = 1 (f = x³ + yz: order ≥ 2 only at the
origin, which has order 2; order 3 nowhere). Exercise 4: the union of the four coordinate
hyperplanes of 𝔸⁴, whose strata of constant order j are the points with exactly j vanishing
coordinates: the origin has order 4, (0, 0, 1, 1) order 2, (1, 1, 1, 1) order 0; and
Hauser's Y = V(xᵏyˡzᵐwⁿ) at the origin has order k + l + m + n. The points are the
K-rational points ratPoint a, K a field of characteristic zero.