# Coordinator check (harvest of ai/2026-10-23-hironaka-char-p-resolution-review).
# Question: for E_k = (x^2 + y^k, 2) in char 2 (k odd), is x in wp(E_k,1)
# under the manuscript's algebraic definition (p.17, Thm 4.4)?
# The degree-2 Veronese of A = O[J T^2, Diff^(1)(J) T] is the Rees algebra of
# I = J + (Diff^(1) J)^2 = (f, y^(2k-2)); so h T^2 is integral over A iff h is
# in the integral closure of I, and x T is integral iff x^2 is.
R.<x,y> = PolynomialRing(GF(2))
for k in [3,5,7]:
    f = x^2 + y^k
    I = R.ideal([f, y^(2*k-2)])
    Ibar = R.ideal(I.integral_closure())
    print("k =", k, " I =", I.gens(), " integral closure gens:", Ibar.gens())
    print("   y^k in closure:", y^k in Ibar, "  x^2 in closure:", x^2 in Ibar,
          "  x*y^(k-2)... (x y^((k-1)/2)) in closure:", x*y^((k-1)//2) in Ibar)