Hasse-Schmidt derivatives 2.0

docs/oscar_references.bib

@@ -1067,6 +1067,58 @@ @Book{Ful98
   doi           = {10.1007/978-1-4612-1700-8}
 }
 
+@Article{FKRS21,
+  author        = {Fruehbis-Krueger, Anne and Ristau, Lukas and Schober, Bernd},
+  title         = {Embedded desingularization for arithmetic surfaces -- toward a parallel implementation}, 
+  year          = {2021},
+  pages         = {32},
+  eprint        = {1712.08131},
+  primaryClass  = {math.AG},
+  doi           = {https://doi.org/10.1090/mcom/3624}
+}
+
+@Article{Hasse1937,
+  author        = {Hasse, H.},
+  journal       = {Journal für die reine und angewandte Mathematik},
+  keywords      = {algebraic function fields; higher differential quotients},
+  pages         = {215-223},
+  title         = {Noch eine Begründung der Theorie der höheren Differentialquotienten in einem algebraischen Funktionenkörper einer Unbestimmten. (Nach einer brieflichen Mitteilung von F.K. Schmidt in Jena).},
+  url           = {http://eudml.org/doc/150015},
+  volume        = {177},
+  year          = {1937}
+}
+
+@Book{Cut04,
+  title         = {Resolution of Singularities},
+  author        = {Cutkosky, S.D.},
+  isbn          = {9780821872383},
+  series        = {Graduate studies in mathematics},
+  url           = {https://books.google.de/books?id=OkAppJ7dXsgC},
+  publisher     = {American Mathematical Soc.}
+}
+
+@Article{Haze11,
+  author        = {Hazewinkel, Michiel},
+  title         = {Hasse-Schmidt Derivations and the Hopf Algebra of Non-Commutative Symmetric Functions},
+  journal       = {Axioms},
+  volume        = {1},
+  year          = {2012},
+  number        = {2},
+  pages         = {149--154},
+  issn          = {2075-1680},
+  doi           = {10.3390/axioms1020149}
+}
+
+@Misc{Haze11,
+  title         = {Hasse-Schmidt derivations and the Hopf algebra of noncommutative symmetric functions}, 
+  author        = {Michiel Hazewinkel},
+  year          = {2011},
+  eprint        = {1110.6108},
+  archivePrefix = {arXiv},
+  primaryClass  = {math.RA},
+  url           = {https://arxiv.org/abs/1110.6108}
+}
+
 @Article{GH12,
   author        = {Grimm, Thomas W. and Hayashi, Hirotaka},
   title         = {F-theory fluxes, chirality and Chern-Simons theories},

experimental/HasseSchmidt/src/HasseSchmidt.jl

@@ -0,0 +1,162 @@
+export hasse_derivatives
+
+### We consider Hasse-Schmidt derivatives of polynomials as seen in
+###
+###   	[FKRS21](@cite) Fruehbis-Krueger, Ristau, Schober: 'Embedded desingularization for arithmetic surfaces -- toward a parallel implementation'
+###			
+### This is a special case of a more general definition of a Hasse-Schmidt derivative. These more general and rigorous definitions can be found in the following sources:
+###
+###			[Cut04](@cite) Cutkosky: 'Resolution of Singularities'
+###			[Haze11](@cite) Michiel Hazewinkel: 'Hasse-Schmidt derivations and the Hopf algebra of noncommutative symmetric functions'
+###
+
+################################################################################
+### HASSE-SCHMIDT derivatives for single polynomials
+
+@doc raw"""
+    hasse_derivatives(f::MPolyRingElem)
+
+Return a list of Hasse-Schmidt derivatives of `f`, each with a multiindex `[a_1, ..., a_n]`, where `a_i` describes the number of times `f` was derived w.r.t. the `i`-th variable. 
+
+Hasse-Schmidt derivatives as seen in [FKRS21](@cite). 
+For more general and rigorous definition see [Cut04](@cite) or [Haze11](@cite).
+
+# Examples
+```jldoctest
+julia> R, (x, y) = polynomial_ring(ZZ, ["x", "y"]);
+
+julia> f = 5*x^2 + 3*y^5;
+
+julia> hasse_derivatives(f)
+8-element Vector{Vector{Any}}:
+ [[0, 5], 3]
+ [[0, 4], 15*y]
+ [[0, 3], 30*y^2]
+ [[2, 0], 5]
+ [[0, 2], 30*y^3]
+ [[1, 0], 10*x]
+ [[0, 1], 15*y^4]
+ [[0, 0], 5*x^2 + 3*y^5]
+```
+"""
+function hasse_derivatives(f::MPolyRingElem)
+  R = parent(f)
+  # x = gens(R)
+  # n = ngens(R)
+  Rtemp, t = polynomial_ring(R, :t => 1:ngens(R))
+  F = evaluate(f, gens(R) + t)
+  return [[degrees(monomial(term, 1)), coeff(term, 1)] for term in terms(F)]
+end
+
+function hasse_derivatives(f::MPolyQuoRingElem)
+  error("Not implemented. For experts, however, there is an internal function called _hasse_derivatives, which works for elements of type MPolyQuoRingElem")
+end
+
+function hasse_derivatives(f::Oscar.MPolyLocRingElem)
+  error("Not implemented. For experts, however, there is an internal function called _hasse_derivatives, which works for elements of type Oscar.MPolyLocRingElem")
+end
+
+function hasse_derivatives(f::Oscar.MPolyQuoLocRingElem)
+  error("Not implemented. For experts, however, there is an internal function called _hasse_derivatives, which works for elements of type Oscar.MPolyQuoLocRingElem")
+end
+
+
+
+
+################################################################################
+### internal functions for expert use
+
+# MPolyQuoRingElem (internal, expert use only)
+@doc raw"""
+    _hasse_derivatives(f::MPolyQuoRingElem)
+
+Return a list of Hasse-Schmidt derivatives of lift of `f`, each with a multiindex `[a_1, ..., a_n]`, where `a_i` describes the number of times `f` was derived w.r.t. the `i`-th variable.
+
+# Examples
+```jldoctest
+julia> R, (x, y) = polynomial_ring(ZZ, ["x", "y"]);
+
+julia> I = ideal(R, [x - 1]);
+
+julia> RQ, phi = quo(R, I);
+
+julia> f = phi(2*y^4);
+
+julia> Oscar._hasse_derivatives(f)
+5-element Vector{Vector{Any}}:
+ [[0, 4], 2]
+ [[0, 3], 8*y]
+ [[0, 2], 12*y^2]
+ [[0, 1], 8*y^3]
+ [[0, 0], 2*y^4]
+```
+"""
+function _hasse_derivatives(f::MPolyQuoRingElem)
+  return hasse_derivatives(lift(f)) 
+end
+
+# Oscar.MPolyLocRingElem (internal, expert use only)
+@doc raw"""
+    _hasse_derivatives(f::Oscar.MPolyLocRingElem)
+
+Return a list of Hasse-Schmidt derivatives of numerator of `f`, each with a multiindex `[a_1, ..., a_n]`, where `a_i` describes the number of times `f` was derived w.r.t. the `i`-th variable.
+
+# Examples
+```jldoctest
+julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"]);
+
+julia> m = ideal(R, [x - 3, y - 2, z + 1]);
+
+julia> U = complement_of_prime_ideal(m);
+
+julia> Rloc, phi = localization(R, U);
+
+julia> f = phi(2*z^5);
+
+julia> Oscar._hasse_derivatives(f)
+6-element Vector{Vector{Any}}:
+ [[0, 0, 5], 2]
+ [[0, 0, 4], 10*z]
+ [[0, 0, 3], 20*z^2]
+ [[0, 0, 2], 20*z^3]
+ [[0, 0, 1], 10*z^4]
+ [[0, 0, 0], 2*z^5]
+```
+"""
+function _hasse_derivatives(f::Oscar.MPolyLocRingElem)
+  return hasse_derivatives(numerator(f)) 
+end
+
+# Oscar.MPolyQuoLocRingElem (internal, expert use only)
+@doc raw"""
+    _hasse_derivatives(f::Oscar.MPolyQuoLocRingElem)
+
+Return a list of Hasse-Schmidt derivatives of lifted numerator of `f`, each with a multiindex `[a_1, ..., a_n]`, where `a_i` describes the number of times `f` was derived w.r.t. the `i`-th variable.
+
+# Examples
+```jldoctest
+julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"]);
+
+julia> I = ideal(R, [x^3 - 1]);
+
+julia> RQ, phi = quo(R, I);
+
+julia> p = ideal(R, [z]);
+
+julia> U = complement_of_prime_ideal(p);
+
+julia> RQL, iota = localization(RQ, U);
+
+julia> f = iota(phi(4*y^3));
+
+julia> Oscar._hasse_derivatives(f)
+4-element Vector{Vector{Any}}:
+ [[0, 3, 0], 4]
+ [[0, 2, 0], 12*y]
+ [[0, 1, 0], 12*y^2]
+ [[0, 0, 0], 4*y^3]
+```
+"""
+function _hasse_derivatives(f::Oscar.MPolyQuoLocRingElem)
+  return hasse_derivatives(lifted_numerator(f))
+end

experimental/HasseSchmidt/test/runtests.jl

@@ -0,0 +1,117 @@
+@testset "hasse_derivatives" begin
+  R, (x, y) = polynomial_ring(ZZ, ["x", "y"]);
+
+  result_a1 = [ [[3, 0], 1],
+                [[2, 0], 3*x],
+                [[1, 0], 3*x^2],
+                [[0, 0], x^3]]
+  @test result_a1 == hasse_derivatives(x^3)
+
+  result_a2 = [ [[0, 5], 3],
+                [[0, 4], 15*y],
+                [[0, 3], 30*y^2],
+                [[2, 0], 5],
+                [[0, 2], 30*y^3],
+                [[1, 0], 10*x],
+                [[0, 1], 15*y^4],
+                [[0, 0], 5*x^2 + 3*y^5]]
+  @test result_a2 == hasse_derivatives(5*x^2 + 3*y^5)
+
+  result_a3 = [ [[2, 3], 1],
+                [[2, 2], 3*y],
+                [[1, 3], 2*x],
+                [[2, 1], 3*y^2],
+                [[1, 2], 6*x*y],
+                [[0, 3], x^2],
+                [[2, 0], y^3],
+                [[1, 1], 6*x*y^2],
+                [[0, 2], 3*x^2*y],
+                [[1, 0], 2*x*y^3],
+                [[0, 1], 3*x^2*y^2],
+                [[0, 0], x^2*y^3]]
+  @test result_a3 == hasse_derivatives(x^2*y^3)
+
+  result_a4 = [ [[4, 0], 1],
+                [[3, 0], 4*x],
+                [[2, 0], 6*x^2],
+                [[0, 2], 1],
+                [[1, 0], 4*x^3],
+                [[0, 1], 2*y],
+                [[0, 0], x^4 + y^2]]
+  @test result_a4 == hasse_derivatives(x^4 + y^2)
+
+  result_a5 = [ [[2, 1], 1],
+                [[1, 2], 1],
+                [[2, 0], y],
+                [[1, 1], 2*x + 2*y],
+                [[0, 2], x],
+                [[1, 0], 2*x*y + y^2],
+                [[0, 1], x^2 + 2*x*y],
+                [[0, 0], x^2*y + x*y^2]]
+  @test result_a5 == hasse_derivatives(x^2*y + x*y^2)
+end
+
+@testset "hasse_derivatives finite fields" begin
+  R, (x, y, z) = polynomial_ring(GF(3), ["x", "y", "z"]);
+
+  result_b1 = [ [[2, 0, 0], 1],
+                [[0, 2, 0], 1],
+                [[1, 0, 0], 2*x],
+                [[0, 1, 0], 2*y],
+                [[0, 0, 0], x^2 + y^2]]
+  @test result_b1 == hasse_derivatives(x^2 + y^2)
+
+  result_b2 = [ [[0, 0, 6], 1],
+                [[2, 1, 0], 1],
+                [[0, 0, 3], 2*z^3],
+                [[2, 0, 0], y],
+                [[1, 1, 0], 2*x],
+                [[1, 0, 0], 2*x*y],
+                [[0, 1, 0], x^2],
+                [[0, 0, 0], x^2*y + z^6]]
+  @test result_b2 == hasse_derivatives(x^2*y + z^6)
+end
+
+@testset "_hasse_derivatives MPolyQuoRingElem" begin
+  R, (x, y, z) = polynomial_ring(ZZ, ["x", "y", "z"]);
+  I = ideal(R, [x^2 - 1]);
+  RQ, _ = quo(R, I);
+
+  result_c1 = [ [[0, 4, 0], 3],
+                [[0, 3, 0], 12*y],
+                [[0, 2, 0], 18*y^2],
+                [[0, 1, 0], 12*y^3],
+                [[0, 0, 0], 3*y^4]]
+  @test result_c1 == Oscar._hasse_derivatives(RQ(3y^4))
+end
+
+@testset "_hasse_derivatives Oscar.MPolyLocRingElem" begin
+  R, (x, y, z) = polynomial_ring(ZZ, ["x", "y", "z"]); 
+  m = ideal(R, [x, y, z]); # max ideal
+  U = complement_of_prime_ideal(m);
+  RL, _ = localization(R, U);
+
+  result_d1 = [ [[3, 0, 0], 5],
+                [[2, 0, 0], 15*x],
+                [[1, 0, 0], 15*x^2],
+                [[0, 0, 0], 5*x^3]]
+  @test result_d1 == Oscar._hasse_derivatives(RL(5x^3))
+end
+
+@testset "_hasse_derivatives Oscar.MPolyQuoLocRingElem" begin
+  R, (x, y, z) = polynomial_ring(ZZ, ["x", "y", "z"]); 
+  I = ideal(R, [x^2 - 1]);
+  RQ, _ = quo(R, I);
+  m = ideal(R, [x, y, z]); # max ideal
+  U = complement_of_prime_ideal(m);
+  RQL, _ = localization(RQ, U);
+
+  result_e1 = [	[[0, 0, 5], 2],
+                [[0, 0, 4], 10*z],
+                [[0, 0, 3], 20*z^2],
+                [[0, 0, 2], 20*z^3],
+                [[0, 0, 1], 10*z^4],
+                [[0, 0, 0], 2*z^5]]
+  @test result_e1 == Oscar._hasse_derivatives(RQL(2z^5))
+end
+