nc-res-computation.hpp

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/* Copyright 2014-2021, Michael E. Stillman */
 
#ifndef _nc_res_computation_hpp_
#define _nc_res_computation_hpp_

 
#include "comp-res.hpp"
#include "NCAlgebras/FreeAlgebraQuotient.hpp"
#include "matrix.hpp"
#include "matrix-con.hpp"

class NCResComputation : public ResolutionComputation
{
 private:
  NCResComputation(const FreeAlgebraQuotient& ring,
                   const Matrix& gbIdealMatrix,
                   int max_level);
 
 public:
  friend ResolutionComputation* createNCRes(const Matrix* groebnerBasisMatrix,
                                            int max_level,
                                            int strategy);
 
  virtual ~NCResComputation() {}
 
 protected:
  // These functions override those in ResolutionComputation
  bool stop_conditions_ok()
  {
    // We ignore all stopping conditions except length_limit, degree_limit
    return true;
  }
 
  void start_computation() { 
    std::cout << "Starting computation." << std::endl;
  }
 
  int complete_thru_degree() const { return 0; }
  // The computation is complete up through this slanted degree.
 
  const Matrix /* or null */* get_matrix(int level) {
    if (level == 1) return &mInputModuleGB;
    MatrixConstructor matCon(get_free(level-1), get_free(level));
    return matCon.to_matrix();
  }
 
  MutableMatrix /* or null */* get_matrix(int slanted_degree, int level)
  {
    (void) slanted_degree;
    (void) level;
    return nullptr;
  }
 
  const FreeModule /* or null */* get_free(int level) {
    if (level == 0) return mInputModuleGB.rows();
    if (level == 1) return mInputModuleGB.cols();
    return mInputModuleGB.get_ring()->make_FreeModule(0);
  }
 
  M2_arrayint get_betti(int type) const
  {
    (void) type;
    return nullptr;
  }
  // type is documented under rawResolutionBetti, in engine.h
 
  void text_out(buffer& o) const {
    o << "Noncommutative resolution";
  }
 
 private:
  // input information coming from M2 objects
  //const Matrix& mInputIdealGB; // probably a computation type from a partial GB.
  const Matrix& mInputModuleGB;  // maybe not a GB of the module either...
};
 
#if 0
//  Data Types
class NCSchreyerResolution
{
  // internal information for this resolution computation
  std::vector<Level> mLevels;
  MemoryBlock mMemoryBlock; // where all the monomials are stored
  std::unordered_set<Monom> mAllMonomials; // or <int*>?
};
 
class Level
{
  std::vector<Element> mElements;
  // SchreyerOrder at this level
};
 
class Element
{
  ModulePoly mPoly;
  // degree
  // component of previous step
  // bool indicating that the element has been computed entirely?
};
 
class ModulePoly
{
  ElementArray mElementArray;
  std::vector<int> mComponents;
  std::vector<int> mMonomials;
  // above could be std::vector of Monoms, or maybe even together
  // a std::vector of a struct.
};
 
#endif
 
// to create the frame:
 
// 1. Fill in level 0 - (Take the degree information from the target mInputModGB)
// 2. Fill in level 1 - (This is really just the info on the entries of mInputModGB)
//                      This is more difficult the GB is not known out to that point yet.
// 3. Fill in level 2 - This requires us to find all the spairs on module elements vs
//                      elements of the ring Groebner basis
// 4. Rinse and repeat.
 
// detailed notes on a basic example
// consider the right ideal (yyx,yxx)R with R = QQ<|x,y|>/(x*y - y*x)
 
// Step 1:
// level 0: right module generated by e_1
 
// Step 2:
// level 1: f_1 --> e_1yyx, f_2 --> e_1yxx
 
// other gb elements? (e_1yyx) * y - (e_1 yy)*(x*y - y*x) = e_1 yyyx
//                    (e_1yxx) * y - (e_1 yx)*(x*y - y*x) = e_1 yxyx ~> e_1yyxx - (e_1yyx)*x = 0
 
// f_3 --> e_1 yyyx, etc (another f_j for each e_i y^j x for j >= 3)
 
// Level 2: 
 
// g_1 --> f_1.y = e_1.yyx.y  <~~> e_1.yyxy = e_1.yyyx, now have to add f_3 to level 1
 
// g_1 --> f_1.y = e_1.yyx.y  <~~> f_1.y - f_3   Claim: this element maps to zero under the previous map
// g_1 will have label: e_1.yyx.y      e_1yyxy - e_1yyyx = e_1yy.(xy - yx) = 0
 
// g_2 --> f_2.y = e_1.yxx.y  <~~> f_2.y - f_1.x    e_1yxxy - e1yyxx = 0
// g_2 will have label: e_1.yxx.y
 
// g_3 --> f_3.y = e_1.yyyx.y <~~> f_3.y - f_4  (etc)
// g_3 will have label: e_1.yyyx.y
 
// for a term, we have:  label - product of all leadTerms all the way back to level 0, e.g. e_1.yyyx.y (without the dots)
 
// to compare
// g_1.m_1; g_1 = f_1.v_1
// g_2.m_2; g_2 = f_2.v_2, f_1 and f_2 are further back etc
 
// first compare total monomials of g_1.m_1 and g_2.m_2. if there is a winner, take that term.
// if there is a tie at this stage, 
 
// example:
// level 0: e_1
// level 1: f_1 = e_1.xy, f_2 = e_1.y^3, f_3 = e_1.yx
// level 2: g_1 = f_1.xyy g_2 = f_1.xxy, g_3 = f_2.x
 
// How to compare f_i.m and f_j.n:  f_1.xxx vs f_2.xy
//             labels are e_1.xyxxx  e_1.yyyxy (compare here).  One is a winner.
 
// Level 3:
// we are done, because:
//  1. All lead terms of images of g's are in different f components, and
//  2. There are no overlaps with these lead terms with the ring.
 
// level degree table:
// degrees rows, levels columns (betti diagram in M2)
// (entries indicate order of computation)
 
// entries with the same letter may be computed at the same time
 
/*
    0     1     2     3        0    1    2    3
0   a     .     .     .                  pb   pc (or *during*)
1   .     b     c     d
2   .     c     d
3   .     d
*/
 
// creation of the frame
// creation of level0and1 (translation of matrix to our information)
// fillInSyzygies - fill in a level and degree of the frame.
// F4 matrix creation and reduction code
// take result in the frame and translate back
ResolutionComputation* createNCRes(const Matrix* m,
                                   int max_level,
                                   int strategy);
 
#endif
 
// Local Variables:
// compile-command: "make -C $M2BUILDDIR/Macaulay2/e "
// indent-tabs-mode: nil
// End: