Points.m2

-- -*- coding: utf-8 -*-
newPackage(
	"Points",
    	Version => "3.0", 
    	Date => "29 June 2008, revised by DE June 2016, revised by FG and JWS June 2018",
    	Authors => {
	     {Name => "Mike Stillman", Email => "mike@math.cornell.edu", HomePage => "http://www.math.uiuc.edu/Macaulay2/"},
     	     {Name => "Gregory G. Smith", Email => "ggsmith@mast.queensu.ca"},
	     {Name => "Stein A. Strømme", Email => "stromme@math.uib.no"},
	     {Name => "David Eisenbud", Email => "de@msri.org"},
	     {Name => "Federico Galetto", Email => "galetto.federico@gmail.com", HomePage => "http://math.galetto.org"},
	     {Name => "Joseph W. Skelton", Email => "jskelton@tulane.edu"}
	     },
    	Headline => "sets of points",
	Keywords => {"Examples and Random Objects"},
	PackageExports => {"LexIdeals"},
    	DebuggingMode => false
    	)

export {
--   Points in affine space
     "affinePointsMat",
     "affinePoints",
     "affinePointsByIntersection",
     "affineMakeRingMaps",
---------     
--    points in projective space
     "randomPointsMat",
     "AllRandom",
     "points",
     "randomPoints",
     "omegaPoints",
     "expectedBetti",
     "minMaxResolution",
---------------------------------------------------------------------
-- FG: fat points, and new projective points (v3)
---------------------------------------------------------------------
     "affineFatPoints",
     "affineFatPointsByIntersection",
     "projectivePoints",
     "VerifyPoints",
     "projectivePointsByIntersection",
     "projectiveFatPointsByIntersection",
     "projectiveFatPoints"
     }

///
restart
loadPackage("Points", Reload=>true)
randomPointsMat
omegaPoints
///

affineMakeRingMaps = method (TypicalValue => List)
affineMakeRingMaps (Matrix, Ring) := List => (M,R) -> (
     K := coefficientRing R;
     pts := entries transpose M;
     apply(pts, p -> map(K, R, p))
     )

addNewMonomial = (M,col,monom,maps) -> (
     -- M is an s by s+1 matrix, s=#points
     -- monom is a monomial
     -- maps is a list of s ring maps, which will give the values
     --  of the monom at the points
     -- replaces the 'col' column of M with the values of monom
     --    at the s points.
     scan(#maps, i -> M_(i,col) = maps#i monom)
     )

affinePointsByIntersection = method(TypicalValue => List)
affinePointsByIntersection (Matrix,Ring) := (M,R) -> (
     flatten entries gens gb intersect apply (
       entries transpose M, p -> ideal apply(#p, i -> R_i - p#i)))

reduceColumn = (M,Mchange,H,c) -> (
     -- M is a mutable matrix
     -- Mchange is either null, or a matrix with same number of columns as M
     -- H is a hash table: H#r == c if column c has pivot for row r 
     -- returns true if the element reduces to 0
     r := numRows M - 1;
     while r >= 0 do (
	  a := M_(r,c);
	  if a != 0 then (
	       -- is there a pivot?
	       if not H#?r then (
		    b := 1/a; -- was 1//a
		    columnMult(M, c, b);
		    if Mchange =!= null then columnMult(Mchange, c, b);
		    H#r = c;
		    return false;
		    )
	       else (
	       	    pivotc := H#r;
	       	    columnAdd(M, c, -a, pivotc);
		    if Mchange =!= null then columnAdd(Mchange, c, -a, pivotc);
	       ));
     	  r = r-1;
	  );
     true
     )

affinePointsMat = method()
affinePointsMat(Matrix,Ring) := (M,R) -> (
     -- The columns of M form the points.  M should be a matrix of size
     -- n by s, where n is the number of variables of R
     --
     K := coefficientRing R;
     s := numgens source M;
     -- The local data structures:
     -- (P,PC) is the matrix which contains the elements to be reduced
     -- Fs is used to evaluate monomials at the points
     -- H is a hash table used in Gaussian elimination: it contains the
     --    pivot columns for each row
     -- L is the sum of monomials which is still to be done
     -- Lhash is a hashtable: Lhash#monom = i means that only 
     --    R_i*monom, ..., R_n*monom should be considered
     -- G is a list of GB elements
     -- inG is the ideal of initial monomials for the GB
     Fs := affineMakeRingMaps(M,R);
     P := mutableMatrix map(K^s, K^(s+1), 0);
     H := new MutableHashTable; -- used in the column reduction step
     Lhash := new MutableHashTable; -- used to determine which monomials come next
     L := 1_R;
     Lhash#L = 0; -- start with multiplication by R_0
     thiscol := 0;
     inG := trim ideal(0_R);
     inGB := forceGB gens inG;
     Q := {}; -- the list of standard monomials
     --ntimes := 0;
     while (L = L % inGB) != 0 do (
	  --ntimes = ntimes + 1;
	  --if #Q === s then print "got a basis";
	  --print("size of L = "| size(L));
	  -- First step: get the monomial to consider
	  monom := someTerms(L,-1,1);
	  L = L - monom;
	  -- Now fix up the matrix P
          addNewMonomial(P,thiscol,monom,Fs);
          isLT := reduceColumn(P,null,H,thiscol);
	  if isLT then (
	       -- we add to G, inG
	       inG = inG + ideal(monom);
	       inGB = forceGB gens inG;
	       )
	  else (
	       -- we modify L, Lhash, thiscol, and also PC
	       Q = append(Q, monom);
	       L = L + sum apply(toList(Lhash#monom .. numgens R - 1), i -> (
			 newmon := monom * R_i;
			 Lhash#newmon = i;
			 newmon));
	       thiscol = thiscol + 1;
	       )
	  );
     --print("ntimes "|ntimes|" std+inG "|#Q + numgens inG);
     stds := transpose matrix{Q};
     A := transpose matrix{apply(Fs, f -> f stds)};
     (A, stds)
     )

affinePoints = method()
affinePoints (Matrix,Ring) := (M,R) -> (
     -- The columns of M form the points.  M should be a matrix of size
     -- n by s, where n is the number of variables of R
     K := coefficientRing R;
     s := numgens source M;
     -- The local data structures:
     -- (P,PC) is the matrix which contains the elements to be reduced
     -- Fs is used to evaluate monomials at the points
     -- H is a hash table used in Gaussian elimination: it contains the
     --    pivot columns for each row
     -- L is the sum of monomials which is still to be done
     -- Lhash is a hashtable: Lhash#monom = i means that only 
     --    R_i*monom, ..., R_n*monom should be considered
     -- G is a list of GB elements
     -- inG is the ideal of initial monomials for the GB
     Fs := affineMakeRingMaps(M,R);
     P := mutableMatrix map(K^s, K^(s+1), 0);
     PC := mutableMatrix map(K^(s+1), K^(s+1), 0);
     for i from 0 to s-1 do PC_(i,i) = 1_K;
     H := new MutableHashTable; -- used in the column reduction step
     Lhash := new MutableHashTable; -- used to determine which monomials come next
     L := 1_R;
     Lhash#L = 0; -- start with multiplication by R_0
     thiscol := 0;
     G := {};
     inG := trim ideal(0_R);
     inGB := forceGB gens inG;
     Q := {}; -- the list of standard monomials
     nL := 1;
     while L != 0 do (
	  -- First step: get the monomial to consider
	  L = L % inGB;
	  monom := someTerms(L,-1,1);
	  L = L - monom;
	  -- Now fix up the matrices P, PC
          addNewMonomial(P,thiscol,monom,Fs);
	  columnMult(PC, thiscol, 0_K);
	  PC_(thiscol,thiscol) = 1_K;
          isLT := reduceColumn(P,PC,H,thiscol);
	  if isLT then (
	       -- we add to G, inG
	       inG = inG + ideal(monom);
	       inGB = forceGB gens inG;
	       g := sum apply(toList(0..thiscol-1), i -> PC_(i,thiscol) * Q_i);
	       G = append(G, PC_(thiscol,thiscol) * monom + g);
	       )
	  else (
	       -- we modify L, Lhash, thiscol, and also PC
	       Q = append(Q, monom);
	       f := sum apply(toList(Lhash#monom .. numgens R - 1), i -> (
			 newmon := monom * R_i;
			 Lhash#newmon = i;
			 newmon));
	       nL = nL + size(f);
	       L = L + f;
	       thiscol = thiscol + 1;
	       )
	  );
--     print("number of monomials considered = "|nL);
     (Q,inG,G)
     )

---------------------------------------------------------------------
-- FG: fat points, and new projective points (v3)
---------------------------------------------------------------------

-- FG: fat points by intersection
-- INPUT: a matrix M whose columns are coordinates of points,
-- a list mults of multiplicities, and a polynomial ring R
-- OUTPUT: gb of the ideal of the fat point scheme
affineFatPointsByIntersection = method(TypicalValue => List)
affineFatPointsByIntersection (Matrix,List,Ring) := (M,mults,R) -> (
     flatten entries gens gb intersect apply (
       entries transpose M, mults,
       (p,m) -> (ideal apply(#p, i -> R_i - p#i))^m))

-- FG: affine Buchberger-Möller algorithm for fat points
-- INPUT: a matrix M whose columns are coordinates of points,
-- a list mults of multiplicities, and a polynomial ring R
-- OUTPUT: a list containing 1) a list of standard monomials (i.e.,
-- monomials forming a basis of the quotient ring), 2) the initial
-- ideal, and 3) the gb of the ideal of the fat point scheme
-- NOTE: the idea is to reuse the Buchberger-Möller algorithm for
-- reduced points, but instead of simply evaluating polynomials at
-- points, their partial derivatives are also evaluated to ensure
-- vanishing. By Zariski-Nagata, this is the desired ideal.
-- This may not be the most efficient strategy. For further ideas,
-- see Abbott, Kreuzer, Robbiano, Computing zero-dimensional schemes,
-- J. Symbolic Comput., doi:10.1016/j.jsc.2004.09.001
-- WARNING: for reduced points (i.e., when mults is a list of 1s)
-- this performs slightly worse than the original function
affineFatPoints = method()
affineFatPoints (Matrix,List,Ring) := (M,mults,R) -> (
     -- obtain all monomials later used for differentiation
     -- sort in increasing order by degree (then monomial order)
     diffops := flatten entries sort basis(0,max mults - 1,R);
     -- this says how many derivatives to use for each point
     cutoffs := apply(mults,m -> sum(m, i -> binomial((dim R)-1+i,i)));
     s := sum cutoffs;
     -- FG: most of the code below is from the affinePoints method
     -- The local data structures:
     -- (P,PC) is the matrix which contains the elements to be reduced
     -- Fs is used to evaluate monomials at the points
     -- H is a hash table used in Gaussian elimination: it contains the
     --    pivot columns for each row
     -- L is the sum of monomials which is still to be done
     -- Lhash is a hashtable: Lhash#monom = i means that only 
     --    R_i*monom, ..., R_n*monom should be considered
     -- G is a list of GB elements
     -- inG is the ideal of initial monomials for the GB
     K := coefficientRing R;
     Fs := affineMakeRingMaps(M,R);
     P := mutableMatrix map(K^s, K^(s+1), 0);
     PC := mutableMatrix map(K^(s+1), K^(s+1), 0);
     for i from 0 to s-1 do PC_(i,i) = 1_K;
     H := new MutableHashTable; -- used in the column reduction step
     Lhash := new MutableHashTable; -- used to determine which monomials come next
     L := 1_R;
     Lhash#L = 0; -- start with multiplication by R_0
     thiscol := 0;
     G := {};
     inG := trim ideal(0_R);
     inGB := forceGB gens inG;
     Q := {}; -- the list of standard monomials
     nL := 1;
     while L != 0 do (
	  -- First step: get the monomial to consider
	  L = L % inGB;
	  monom := someTerms(L,-1,1);
	  L = L - monom;
	  -- Now fix up the matrices P, PC
	  -- FG: old code called another function addNewMonomial
	  -- FG: I include code here to better evaluate derivatives
	  partials := apply(diffops, del -> diff(del,monom));
	  -- FG: evaluate partials at point up to cutoff
	  c := 0;
	  for i to #Fs-1 do (
	      for j to cutoffs_i-1 do (
		  P_(c+j,thiscol) = Fs#i (partials_j);
		  );
	      c = c + cutoffs_i;
	      );
	  -- FG: remaining code is the same as for reduced points
	  columnMult(PC, thiscol, 0_K);
	  PC_(thiscol,thiscol) = 1_K;
          isLT := reduceColumn(P,PC,H,thiscol);
	  if isLT then (
	       -- we add to G, inG
	       inG = inG + ideal(monom);
	       inGB = forceGB gens inG;
	       g := sum apply(toList(0..thiscol-1), i -> PC_(i,thiscol) * Q_i);
	       G = append(G, PC_(thiscol,thiscol) * monom + g);
	       )
	  else (
	       -- we modify L, Lhash, thiscol, and also PC
	       Q = append(Q, monom);
	       f := sum apply(toList(Lhash#monom .. numgens R - 1), i -> (
			 newmon := monom * R_i;
			 Lhash#newmon = i;
			 newmon));
	       nL = nL + size(f);
	       L = L + f;
	       thiscol = thiscol + 1;
	       )
	  );
--     print("number of monomials considered = "|nL);
     (Q,inG,G)
     )


-- FG: Buchberger-Möller for projective points
-- INPUT: a matrix M whose columns are projective coordinates of
-- points, and a polynomial ring R
-- OUTPUT: a list containing 1) the initial ideal,
-- and 2) the gb of the ideal of the set of points
projectivePoints = method(Options => {VerifyPoints => true})
projectivePoints (Matrix,Ring) := opts -> (M,R) -> (
    if opts.VerifyPoints then M = removeBadPoints M;
    -- FG: the code is mostly like the affine case
    -- but now we proceed degree by degree
     K := coefficientRing R;
     s := numgens source M;
     Fs := affineMakeRingMaps(M,R);
     G := {};
     inG := trim ideal(0_R);
     inGB := forceGB gens inG;
     deg := 1;
     while not stoppingCriterion(deg,inG,s) do (
	 L := sum flatten entries basis(deg,R);
	 L = L % inGB;
	 P := mutableMatrix map(K^s, K^(s+1), 0);
	 PC := mutableMatrix map(K^(s+1), K^(s+1), 0);
	 for i from 0 to s-1 do PC_(i,i) = 1_K;
	 H := new MutableHashTable; -- used in the column reduction step
	 thiscol := 0;
	 Q := {}; -- list of standard monomials of current degree
	 while L != 0 do (
	      -- First step: get the monomial to consider
	      monom := someTerms(L,-1,1);
	      L = L - monom;
	      -- Now fix up the matrices P, PC
	      addNewMonomial(P,thiscol,monom,Fs);
	      columnMult(PC, thiscol, 0_K);
	      PC_(thiscol,thiscol) = 1_K;
	      isLT := reduceColumn(P,PC,H,thiscol);
	      if isLT then (
		   -- we add to G, inG
		   inG = inG + ideal(monom);
		   g := sum apply(toList(0..thiscol-1), i -> PC_(i,thiscol) * Q_i);
		   G = append(G, PC_(thiscol,thiscol) * monom + g);
		   )
	      else (
		   -- add to standard monomials
		   Q = append(Q, monom);
		   thiscol = thiscol + 1;
		   )
	      );
	  inGB = forceGB gens inG;
	  -- proceed with next degree
	  deg = deg + 1;
	  );
     (inG,G)
     )

-- FG: stopping criterion for projective BM
-- INPUT: an integer deg for the current degree,
-- a monomial ideal inG (initial ideal of the ideal of points as
-- computed so far), and an integer multPts which is the degree of
-- the point scheme, i.e., the sum of the degrees of all the points
-- OUTPUT: true if the Hilbert function of the initial ideal is
-- equal to the expected degree for the given points (this is when
-- the BM algorithm should stop)
-- TO DO: implement better stopping criterion from Abbot, Kreuzer, Robbiano
stoppingCriterion = (deg,inG,multPts) -> (
    -- if the initial ideal is zero, then continue
    if zero inG then return false else
    -- otherwise stop when multiplicity is attained
    hilbertFunction(deg,inG) == multPts
    )

-- FG: remove zero and duplicate points
-- INPUT: a matrix M whose columns are projective coordinates of
-- points
-- OUTPUT: a matrix obtained from M by removing zero columns and
-- columns that are not scalar multiples of previous columns
-- NOTE: if these points are not removed, the projective BM
-- algorithm above will not terminate!
removeBadPoints = M -> (
    -- remove zero columns
    N := compress M;
    -- remove columns that define same projective points
    lastcol := numColumns(N)-1;
    thiscol := 0;
    while thiscol < lastcol do (
	L := toList(thiscol+1..lastcol);
	dupcols := select(L,i->rank(N_{thiscol,i})<2);
	N = submatrix'(N,dupcols);
	lastcol = lastcol - #dupcols;
	thiscol = thiscol + 1;
	);
    return N;
    )

-- FG: projective points by intersection
-- INPUT: a matrix M whose columns are coordinates of points,
-- and a polynomial ring R
-- OUTPUT: gb of the ideal of the projective points
projectivePointsByIntersection = method(TypicalValue => List)
projectivePointsByIntersection (Matrix,Ring) := (M,R) -> (
     flatten entries gens gb intersect apply (
       entries transpose M,
       p -> (trim minors(2,matrix{gens R,p}))
       )
   )

-- FG: projective fat points by intersection
-- INPUT: a matrix M whose columns are coordinates of points,
-- a list mults of multiplicities for each point,
-- and a polynomial ring R
-- OUTPUT: gb of the ideal of the projective fat point scheme
projectiveFatPointsByIntersection = method(TypicalValue => List)
projectiveFatPointsByIntersection (Matrix,List,Ring) := (M,mults,R) -> (
     flatten entries gens gb intersect apply (
       entries transpose M, mults,
       (p,m) -> ((trim minors(2,matrix{gens R,p}))^m)
       )
   )

-- FG: remove zero and duplicate points
-- INPUT: a matrix M whose columns are projective coordinates of
-- points, and a list mults of multiplicities for those points
-- OUTPUT: a matrix obtained from M by removing zero columns and
-- columns that are not scalar multiples of previous columns,
-- and a list of multiplicities for the points in the new matrix
-- NOTE: if a point appears more than once with different
-- multiplicities, the largest multiplicity is retained
removeBadFatPoints = (M,mults) -> (
    -- remove zero columns and their multiplicities
    lastcol := numColumns(M)-1;
    zeroVec := 0_(target M);
    nonzerocols := {};
    for i to lastcol do (
	if M_i != zeroVec then nonzerocols = append(nonzerocols,i);
	);
    N := submatrix(M,nonzerocols);
    newmults := new MutableList from mults_nonzerocols;
    -- remove columns that define same projective points
    -- and their multiplicities
    lastcol = numColumns(N)-1;
    thiscol := 0;
    while thiscol < lastcol do (
	L := toList(thiscol+1..lastcol);
	dupcols := select(L,i->rank(N_{thiscol,i})<2);
	N = submatrix'(N,dupcols);
	newmults#thiscol = max apply({thiscol}|dupcols,i->newmults#i);
	for i in reverse dupcols do (
	    newmults = drop(newmults,{i,i})
	    );
	lastcol = lastcol - #dupcols;
	thiscol = thiscol + 1;
	);
    return (N,new List from newmults);
    )

-- FG: Buchberger-Möller for projective fat points
-- INPUT: a matrix M whose columns are projective coordinates of
-- points, a list mults of multiplicities for those points,
-- and a polynomial ring R
-- OUTPUT: a list containing 1) the initial ideal,
-- and 2) the gb of the ideal of the set of fat points
-- NOTE: for small sets of points this can perform much worse than
-- simply intersecting. The first example where I saw an advantage
-- (of 1 sec) was for 30 points in P^5 with multiplicities 1,2,3
projectiveFatPoints = method(Options => {VerifyPoints => true})
projectiveFatPoints (Matrix,List,Ring) := opts -> (M,mults,R) -> (
    if opts.VerifyPoints then (M,mults) = removeBadFatPoints (M,mults);
     K := coefficientRing R;
     diffops := flatten entries sort basis(0,max mults - 1,R);
     -- this says how many derivatives to use for each point
     cutoffs := apply(mults,m -> sum(m, i -> binomial((dim R)-1+i,i)));
     s := sum cutoffs;
     Fs := affineMakeRingMaps(M,R);
     G := {};
     inG := trim ideal(0_R);
     inGB := forceGB gens inG;
     deg := 1;
     schemedegree := sum(mults,m -> binomial((dim R)-2+m,m-1));
     while not stoppingCriterion(deg,inG,schemedegree) do (
	 L := sum flatten entries basis(deg,R);
	 L = L % inGB;
	 P := mutableMatrix map(K^s, K^(s+1), 0);
	 PC := mutableMatrix map(K^(s+1), K^(s+1), 0);
	 for i from 0 to s-1 do PC_(i,i) = 1_K;
	 H := new MutableHashTable; -- used in the column reduction step
	 thiscol := 0;
	 Q := {}; -- list of standard monomials of current degree
	 while L != 0 do (
	      -- First step: get the monomial to consider
	      monom := someTerms(L,-1,1);
	      L = L - monom;
	      partials := apply(diffops, del -> diff(del,monom));
	      -- FG: evaluate partials at point up to cutoff
	      c := 0;
	      for i to #Fs-1 do (
	      	  for j to cutoffs_i-1 do (
		      P_(c+j,thiscol) = Fs#i (partials_j);
		      );
	      	  c = c + cutoffs_i;
	      	  );
	      columnMult(PC, thiscol, 0_K);
	      PC_(thiscol,thiscol) = 1_K;
	      isLT := reduceColumn(P,PC,H,thiscol);
	      if isLT then (
		   -- we add to G, inG
		   inG = inG + ideal(monom);
		   g := sum apply(toList(0..thiscol-1), i -> PC_(i,thiscol) * Q_i);
		   G = append(G, PC_(thiscol,thiscol) * monom + g);
		   )
	      else (
		   -- add to standard monomials
		   Q = append(Q, monom);
		   thiscol = thiscol + 1;
		   )
	      );
	  inGB = forceGB gens inG;
	  -- proceed with next degree
	  deg = deg + 1;
	  );
     (inG,G)
     )

---------------------------------------------------------------------
-- FG: end of v3 code
---------------------------------------------------------------------


-----------------Homogeneous codes

randomPointsMat = method(Options =>{AllRandom =>false})
randomPointsMat(Ring, ZZ) := opts -> (R,n) -> (
	d := numgens R;
	if opts.AllRandom == true then return random(R^d, R^n);
	
	m1 := id_(R^d)|transpose matrix(R,{toList(d:1)});
	if n<=d+1 then return m1_(toList(0..n-1));
	
	m3 := random(R^d,R^(n-d-1));
	m1 | m3
	)


points = (pointsmat) -> (
        mm := vars ring pointsmat;
	if rank source mm =!= rank target pointsmat then 
		error "wrong size matrix";
        ids := toSequence apply(rank source pointsmat, 
	    i -> image(mm * (syz transpose pointsmat_{i})));
        ideal intersect ids
	)
    
randomPoints = (r,n) -> (
	x := symbol x;
	R := ZZ/101[x_0 .. x_r];
	pmat := randomPointsMat(R,n);
	points pmat
	)


testPoints = ()->(
    	a := symbol a;
    	b := symbol b;
    	c := symbol c;		
	R := ZZ/101[a,b,c];
	pmat := matrix(R,{{1,0,0},{0,1,0},{0,0,1}});
	assert(points pmat == image matrix(R, {{a*b, a*c, b*c}}))
	)

omegaPoints = (pointsmat) -> (
	dualmat := syz pointsmat;
	s := (rank source dualmat)-1;
	n := rank source pointsmat;
	r := (rank target pointsmat)-1;
	mm := vars ring pointsmat;
	if rank source mm =!= r+1 then
		error "wrong size matrix";
	R := ring pointsmat;
	mult := matrix(R, table(n, n^2,(i,j) -> (
		if j//n==j%n and j%n==i then 1 else 0)));
	prod := mult*((transpose pointsmat)**dualmat);
	mm = (identity (R^{1}))**mm;
	(mm**(identity (R^(s+1))))*(generators kernel prod)
	)
testOmegaPoints = () -> (
	R := ZZ/101[vars(0..6)];
	testmat := random(R^7,R^11);
	-- testmat = matrix(R, {{1,0,1,5},{0,1,1,11}});
	w := omegaPoints(testmat);
	assert(rank source w == 18 and rank target w == 4)
		)


expectedBetti = (r,n) -> (
	e := 1;
	while binomial(r+e,e)<= n do e=e+1;
	d := e-1;
	a:=n-binomial(r+d,d);
	toprow := apply(toList(1..r),i->
		max((binomial(d+i-1,i-1))*(binomial(r+d,d+i))-a*(binomial(r,i-1)),
		0));
	bottomrow := apply(toList(1..r), i->
		max(a*(binomial(r,i))-(binomial(d+i,i))*(binomial(r+d,d+i+1)),
		0));
	top := apply(toList(0..r),i -> (
		if i == 0 then (0,{0},0)=>1 else
		(i,{},d+i) => toprow#(i-1) ));
	bottom := apply(toList(1..r),i -> (
		(i,{},d+i+1) => bottomrow#(i-1) ));
	new BettiTally from join(top, bottom)
	)
///
restart	
loadPackage("Points", Reload =>true)
expectedBetti(3,5)
minMaxResolution(3,5)
r=3;n=5
	R = ZZ/2[x_(0..n-1)];
	expectedBetti(r,n)
	lexIdeal(R,{1,3,1})
	betti resolution 
///

minMaxResolution = (r,n) -> (
	e := 1;
	while binomial(r+e,e)<= n do e=e+1;
	H := apply(e, i -> binomial(r+i-1,i));
	H = append(H,n-binomial(r+e-1,e-1));
	if last H =!= 0 then H = append(H,0);	
	x := symbol x;
	R := ZZ/2[x_0..x_(r-1)];
	<<"min"<<endl<< expectedBetti(r,n)<<endl;
	<<"max"<<endl<<betti resolution lexIdeal(R,H)<<endl;
)

-- First examples where expected resolution fails.
-- December 25, 1995: 
-- Unfortunately VERY slow in this system (the resolution is fast,
-- but the random number generation and intersections are very slow
-- for example the first uses 65 seconds of cpu time on a sparc 10!!
-- June 2016:
-- now first example takes .07 seconds on a mac air.

eg1 := ()->(
	res randomPoints(6,11)
	)
eg2 := ()->(
	res randomPoints(7,12)
	)
eg3 := ()->(
	res randomPoints(8,13)
	)
eg4 := ()->(
	res randomPoints(10,16)
	)

-- The following method should be much better:

eg := (r,n) -> (
--	print expectedBetti(r,n);
	R := ZZ/31991[vars (0..r)];
	w := omegaPoints(randomPointsMat(R,n));
--	betti resolution( w, DegreeLimit => 1)
	)


beginDocumentation()
doc ///
   Key
    Points
   Headline
    A package for making and studying points in affine and projective spaces
   Description
    Text
     The package has routines for points in affine and projective spaces. The affine
     code, some of which uses the Buchberger-Moeller algorithm to more quickly
     compute the ideals of points in affine space,
     was written by Stillman, Smith and Stromme. The projective code was
     written separately by Eisenbud and Popescu.
     
     The purpose of the projective code was to find as many counterexamples
     as possible to the minimal resolution conjecture; it was of use in the 
     research for the paper 
     "Exterior algebra methods for the minimal resolution conjecture",
     by  David Eisenbud, Sorin Popescu, Frank-Olaf Schreyer and Charles Walter
     (Duke Mathematical Journal. 112 (2002), no.2, 379-395.)
     The first few of these counterexamples are:
     (6,11),
     (7,12),
     (8,13),
     (10,16),
     where the first integer denotes the ambient dimension and the second the 
     number of points. Examples are known in every projective space of dimension >=6
     except for P^9.
     
     In version 3.0, F. Galetto and J.W. Skelton added code to
     compute ideals of fat points and projective points using
     the Buchberger-Moeller algorithm.
///

--documentation for the code for points in projective space
doc ///
   Key
    randomPoints
   Headline
    ideal of a random set of points
   Usage
    i = randomPoints(r,n)
   Inputs
    r:ZZ
     ambient dimension
    n:ZZ
     number of points
   Outputs
    i:Ideal
     ideal of the random points
   Description
    Text
     The script defines a ring R with r+1 variables, and calls
     points(R,randomPointsMat(R, n))
    Example
     betti res randomPoints(11,5)
   SeeAlso
    randomPointsMat
    points
///


doc ///
   Key
    expectedBetti
   Headline
    The betti table of r points in Pn according to the minimal resolution conjecture
   Usage
    L = expectedBetti(r,n)
   Inputs
    r:ZZ
     ambient dimension
    n:ZZ
     number of points
   Outputs
    L:List
   Description
    Text
     The output is the smallest conceivable betti table for a set of
     r points in P^n, which is predicted (incorrectly) by the minimal resolution conjecture.
    Example
     expectedBetti(11,5)
   Caveat
    The MRC is false, so these are sometimes not the actual betti numbers.
   SeeAlso
    expectedBetti
    minMaxResolution
///

doc ///
   Key
    minMaxResolution
   Headline
    Min and max conceivable Betti tables for generic points
   Usage
    minMaxResolution(r,n)
   Inputs
    r:ZZ
     ambient dimension
    n:ZZ
     number of points
   Description
    Text
     prints betti tables corresponding to the minimal resolution conjecture
     and to the lex ideal with the same hilbert function
    Example
     minMaxResolution(3,5)
   SeeAlso
    expectedBetti
///

doc ///
   Key
    omegaPoints
   Headline
    linear part of the presentation of canonical module of points
   Usage
    m = omegaPoints pointsmat
   Inputs
    pointsmat:Matrix
     matrix of ZZ, representing a set of points
   Outputs
    m:Matrix
     linear part of the presentation matrix of the canonical module
   Description
    Text
     given an r+1 x n matrix
     over a ring with r+1 variables, interpreted as a set of 
     n points in P^r, the script produces the linear part
     of the presentation matrix of w_{>=-1}, where w is the
     canonical module of the cone over the points.  It is
     necessary for this to assume that no subset of n+1
     of the points is linearly dependent.  The presentation
     is actually a presentation of w if the points do not
     lie on a rational normal curve (so there are no
     quadratic relations on w_{>=-1}) and impose independent
     conditions on quadrics (so the homogeneous coordinate
     ring is 3-regular, and w is generated in degree -1.
    Example
     R = ZZ/101[vars(0..4)]
     p = randomPointsMat(R,11)
     w = omegaPoints p
     degree (R^1/(points p))
     degree coker w
     betti res (R^1/(points p))
     betti res coker w
   SeeAlso
///


doc ///
   Key
    randomPointsMat
    (randomPointsMat, Ring, ZZ)
    [randomPointsMat, AllRandom]
   Headline
    matrix of homogeneous coordinates of random points in projective space
   Usage
    m=randomPointsMat(R,n)    
   Inputs
    R:Ring
     homogeneous coordinate ring of projective space Pm
    n:ZZ
     number of points
   Outputs
    m:Matrix
     of ZZ
   Description
    Text
     Produces a random m+1 x n matrix of scalars, with columns representing the coordinates
     of the point. The first m+1 x m+1 submatrix is the identity.
    Example
     R = ZZ/31991[vars(0..3)]
     randomPointsMat(R,3)
     randomPointsMat(R,3, AllRandom=>true)
     randomPointsMat(R,7)
///


doc ///
   Key
    points
   Headline
    make the ideal of a set of points
   Usage
    i = points pointsMat
   Inputs
    pointsMat:Matrix
     matrix whose columns are the homogeneous cooredinates of the points
   Outputs
    i:Ideal
   Description
    Text
    Example
     R = ZZ/101[vars(0..4)]
     pointsMat = randomPointsMat(R,11)
     points pointsMat
   SeeAlso
    randomPointsMat
///

doc ///
   Key
    AllRandom
   Headline
    Option to randomPointsMat.
   Description
    Text
     Default is false, in which case the first (up to) r+1 points
     returned are the standard simplex; if true, all the points are random.
   SeeAlso
    randomPointsMat
///


---documentation for the affine code:
document {
     Key => {affineMakeRingMaps, (affineMakeRingMaps,Matrix,Ring)},
     Headline => "evaluation on points",
     Usage => "affineMakeRingMaps(M,R)",
     Inputs => {
     	  "M" => Matrix => "in which each column consists of the coordinates of a point",
	  "R" => PolynomialRing => "coordinate ring of the affine space containing the points",
	  },
     Outputs => {List => "of ring maps corresponding to evaluations at each point"},
     "Giving the coordinates of a point in affine space is equivalent to giving a
     ring map from the polynomial ring to the ground field: evaluation at the point.  Given a
     finite collection of points encoded as the columns of a matrix,
     this function returns a corresponding list of ring maps.",
     EXAMPLE lines ///
     M = random(ZZ^3, ZZ^5)
     R = QQ[x,y,z]
     phi = affineMakeRingMaps(M,R)
     apply (gens(R),r->phi#2 r)
     phi#2
     ///
     }


---the affine code documentation
document {
     Key => {affinePoints, (affinePoints,Matrix,Ring)},
     Headline => "produces the ideal and initial ideal from the coordinates
     of a finite set of points",
     Usage => "(Q,inG,G) = affinePoints(M,R)",
     Inputs => {
     	  "M" => Matrix => "in which each column consists of the coordinates of a point",
	  "R" => PolynomialRing => "coordinate ring of the affine space containing the points",
	  },
     Outputs => {
          "Q" => List => "list of standard monomials",
 	  "inG" => Ideal => "initial ideal of the set of points",
 	  "G" => List => "list of generators for Grobner basis for ideal of points"
 	  },
     "This function uses the Buchberger-Moeller algorithm to compute a grobner basis
     for the ideal of a finite number of points in affine space.  Here is a simple
     example.",
     EXAMPLE lines ///
     M = random(ZZ^3, ZZ^5)
     R = QQ[x,y,z]
     (Q,inG,G) = affinePoints(M,R)
     monomialIdeal G == inG
     ///,
     PARA{},
     "The Buchberger-Moeller algorithm in ",
     TT "points", " may be faster than the alternative method using the intersection
     of the ideals for each point.",
     SeeAlso => {affinePointsByIntersection}
     }

document {
     Key => {affinePointsMat, (affinePointsMat,Matrix,Ring)},
     Headline => "produces the matrix of values of the standard monomials
     on a set of points",
     Usage => "(A,stds) = affinePointsMat(M,R)",
     Inputs => {
     	  "M" => Matrix => "in which each column consists of the coordinates of a point",
	  "R" => PolynomialRing => "coordinate ring of the affine space containing the points",
	  },
     Outputs => {
          "A" => Matrix => "standard monomials evaluated on points",
 	  "stds" => Matrix => "whose entries are the standard monomials",
 	  },
     "This function uses the Buchberger-Moeller algorithm to compute a the matrix ",
     TT "A", " in which the columns are indexed by standard monomials, the rows are
     indexed by points, and the entries are given by evaluation.  The ordering of
     the standard monomials is recorded in the matrix ", TT "stds", " which has a
     single column.
     Here is a simple
     example.",
     EXAMPLE lines ///
     M = random(ZZ^3, ZZ^5)
     R = QQ[x,y,z]
     (A,stds) = affinePointsMat(M,R)
     ///,
     Caveat => "Program does not check that the points are distinct.",
     SeeAlso => {affinePoints},
     }

document {
     Key => {affinePointsByIntersection, (affinePointsByIntersection,Matrix,Ring)},
     Headline => "computes ideal of point set by intersecting maximal ideals",
     Usage => "affinePointsByIntersection(M,R)",
     Inputs => {
     	  "M" => Matrix => "in which each column consists of the coordinates of a point",
	  "R" => PolynomialRing => "coordinate ring of the affine space containing the points",
	  },
     Outputs => {
 	  List => "grobner basis for ideal of a finite set of points",
 	  },
     "This function computes the ideal of a finite set of points by intersecting
     the ideals for each point.  The coordinates of the points are the columns in
     the input matrix ", TT "M", ".",
     EXAMPLE lines ///
     M = random(ZZ^3, ZZ^5)
     R = QQ[x,y,z]
     affinePointsByIntersection(M,R)
     ///,
     SeeAlso => {affinePoints},
     }

---------------------------------------------------------------------
-- FG: documentation for fat points and new projective points 
---------------------------------------------------------------------

doc ///
   Key
    affineFatPoints
    (affineFatPoints,Matrix,List,Ring)
   Headline
    produces the ideal and initial ideal from the coordinates of a finite set of fat points
   Usage
    (Q,inG,G) = affineFatPoints(M,mults,R)
   Inputs
    M:Matrix
     in which each column consists of the coordinates of a point
    mults:List
     in which each element determines the multiplicity of the
     corresponding point
    R:Ring
     coordinate ring of the affine space containing the points
   Outputs
    Q:List
     list of standard monomials
    inG:Ideal
     initial ideal of the set of fat points
    G:List
     list of generators for Grobner basis for ideal of fat points
   Description
    Text
     This function uses a modified Buchberger-Moeller algorithm to
     compute a grobner basis for the ideal of a finite number of
     fat points in affine space.

    Example
     R = QQ[x,y]
     M = transpose matrix{{0,0},{1,1}}
     mults = {3,2}
     (Q,inG,G) = affineFatPoints(M,mults,R)
     monomialIdeal G == inG

    Text
     This algorithm may be faster than
     computing the intersection of the ideals of each fat point.

    Example
     K = ZZ/32003
     R = K[z_1..z_5]
     M = random(K^5,K^12)
     mults = {1,2,3,1,2,3,1,2,3,1,2,3}
     elapsedTime (Q,inG,G) = affineFatPoints(M,mults,R);
     elapsedTime H = affineFatPointsByIntersection(M,mults,R);
     G==H

   Caveat
    For reduced points, this function may be a bit slower than @TO "affinePoints"@.
   SeeAlso
    (affineFatPointsByIntersection,Matrix,List,Ring)
///

doc ///
   Key
    affineFatPointsByIntersection
    (affineFatPointsByIntersection,Matrix,List,Ring)
   Headline
    computes ideal of fat points by intersecting powers of maximal ideals
   Usage
    affineFatPointsByIntersection(M,mults,R)
   Inputs
    M:Matrix
     in which each column consists of the coordinates of a point
    mults:List
     in which each element determines the multiplicity of the
     corresponding point
    R:Ring
     coordinate ring of the affine space containing the points
   Outputs
    :List
     grobner basis for ideal of a finite set of fat points
   Description
    Text
     This function computes the ideal of a finite set of fat points
     by intersecting powers of the maximal ideals of each point.

    Example
     R = QQ[x,y]
     M = transpose matrix{{0,0},{1,1}}
     mults = {3,2}
     affineFatPointsByIntersection(M,mults,R)

   SeeAlso
    (affineFatPoints,Matrix,List,Ring)
///

doc ///
   Key
    projectivePoints
    (projectivePoints,Matrix,Ring)
   Headline
    produces the ideal and initial ideal from the coordinates of a finite set of projective points
   Usage
    (inG,G) = projectivePoints(M,R)
   Inputs
    M:Matrix
     in which each column consists of the projective coordinates of a point
    R:Ring
     homogeneous coordinate ring of the projective space containing the points
   Outputs
    inG:Ideal
     initial ideal of the set of projective points
    G:List
     list of generators for Grobner basis for ideal of projective points
   Description
    Text
     This function uses a modified Buchberger-Moeller algorithm to
     compute a grobner basis for the ideal of a finite number of
     points in projective space.

    Example
     R = QQ[x_0..x_2]
     M = random(ZZ^3,ZZ^5)
     (inG,G) = projectivePoints(M,R)
     monomialIdeal G == inG

    Text
     This algorithm may be faster than
     computing the intersection of the ideals of each projective point.
   Caveat
    This function removes zero columns of @TT "M"@ and duplicate columns giving rise to the same projective point (which prevent the algorithm from terminating). The user can bypass this step with the option @TT "VerifyPoints"@.
   SeeAlso
    (projectivePointsByIntersection,Matrix,Ring)
///

doc ///
   Key
    VerifyPoints
   Headline
    Option to projectivePoints.
   Description
    Text
     Default is true, in which case the function removes zero columns and duplicate columns giving rise to the same projective point.
   SeeAlso
    projectivePoints
///

doc ///
   Key
    [projectivePoints,VerifyPoints]
   Headline
    Option to projectivePoints.
   Description
    Text
     Default is true, in which case the function removes zero columns and duplicate columns giving rise to the same projective point.
   SeeAlso
    projectivePoints
///

doc ///
   Key
    [projectiveFatPoints,VerifyPoints]
   Headline
    Option to projectiveFatPoints.
   Description
    Text
     Default is true, in which case the function removes zero columns and duplicate columns giving rise to the same projective point.
     For duplicate points, a single instance is retained with the largest multiplicity.
   SeeAlso
    projectiveFatPoints
///

doc ///
   Key
    projectivePointsByIntersection
    (projectivePointsByIntersection,Matrix,Ring)
   Headline
    computes ideal of projective points by intersecting point ideals
   Usage
    projectivePointsByIntersection(M,R)
   Inputs
    M:Matrix
     in which each column consists of the projective coordinates of a point
    R:Ring
     homogeneous coordinate ring of the projective space containing the points
   Outputs
    :List
     grobner basis for ideal of a finite set of projective points
   Description
    Text
     This function computes the ideal of a finite set of projective points
     by intersecting the ideals of each point.

    Example
     R = QQ[x,y,z]
     M = transpose matrix{{1,0,0},{0,1,1}}
     projectivePointsByIntersection(M,R)

   SeeAlso
    (projectivePoints,Matrix,Ring)
///

doc ///
   Key
    projectiveFatPointsByIntersection
    (projectiveFatPointsByIntersection,Matrix,List,Ring)
   Headline
    computes ideal of fat points by intersecting powers of point ideals
   Usage
    projectiveFatPointsByIntersection(M,mults,R)
   Inputs
    M:Matrix
     in which each column consists of the projective coordinates of a point
    mults:List
     in which each element determines the multiplicity of the
     corresponding point
    R:Ring
     homogeneous coordinate ring of the projective space containing the points
   Outputs
    :List
     grobner basis for ideal of a finite set of fat points
   Description
    Text
     This function computes the ideal of a finite set of fat points
     by intersecting powers of the ideals of each point.

    Example
     R = QQ[x,y,z]
     M = transpose matrix{{1,0,0},{0,1,1}}
     mults = {3,2}
     projectiveFatPointsByIntersection(M,mults,R)

   SeeAlso
    (projectiveFatPoints,Matrix,List,Ring)
///

doc ///
   Key
    projectiveFatPoints
    (projectiveFatPoints,Matrix,List,Ring)
   Headline
    produces the ideal and initial ideal from the coordinates of a finite set of fat points
   Usage
    (inG,G) = projectiveFatPoints(M,mults,R)
   Inputs
    M:Matrix
     in which each column consists of the projective coordinates of a point
    mults:List
     in which each element determines the multiplicity of the
     corresponding point
    R:Ring
     homogeneous coordinate ring of the projective space containing the points
   Outputs
    inG:Ideal
     initial ideal of the set of fat points
    G:List
     list of generators for Grobner basis for ideal of fat points
   Description
    Text
     This function uses a modified Buchberger-Moeller algorithm to
     compute a grobner basis for the ideal of a finite number of
     fat points in projective space.

    Example
     R = QQ[x,y,z]
     M = transpose matrix{{1,0,0},{0,1,1}}
     mults = {3,2}
     (inG,G) = projectiveFatPoints(M,mults,R)
     monomialIdeal G == inG

   Caveat
    For small sets of points and/or multiplicities, this method might be slower than @TO "projectiveFatPointsByIntersection"@.
   SeeAlso
    (projectiveFatPointsByIntersection,Matrix,List,Ring)
///


TEST///
     M = random(ZZ^3, ZZ^3)
     M = id_(ZZ^3)
     R = QQ[x,y,z]
     (Q,inG,G) = affinePoints(M,R)
     assert( G == {x+y+z-1, z^2-z, y*z, y^2-y})
///
TEST///
setRandomSeed 0
m = randomPoints(3,5) 
R = ring m
assert(m == ideal(
  R_1*R_3-R_2*R_3,R_0*R_3-R_2*R_3,R_1*R_2-R_2*R_3,R_0*R_2-R_2*R_3,R_0*R_1-R_2*R_3)
)
///

TEST///
setRandomSeed 0
R = ZZ/101[a,b,c]
p = randomPointsMat(R,6)
assert(omegaPoints p == R^{1}**matrix {{-36*a+36*b, 19*a-19*b, -30*a+30*c, 19*a-19*c}, {-49*a+b, 0, -24*a+c, 0}, {0,
      32*a+b, 0, 32*a+c}}
)
///

TEST///
assert( 
    expectedBetti(3,5) == new BettiTally from {(0,{0},0) => 1, (1,{},2) => 5, (1,{},3) => 0, (2,{},3) => 5, (2,{},4)
      => 0, (3,{},4) => 0, (3,{},5) => 1}
)
///

TEST///
R = ZZ/11[vars(0..2)]
setRandomSeed 0
assert(
    randomPointsMat(R,5) == R**matrix {{1, 0, 0, 1, -3}, {0, 1, 0, 1, 1}, {0, 0, 1, 1, 3}}
    )
assert(randomPointsMat(R,3,AllRandom=>true) == R**matrix {{-4, 3, -3}, {-3, 3, -3}, {-1, -4, 5}})
///

TEST ///
R = ZZ/32003[vars(0..4), MonomialOrder=>Lex]
M = matrix(ZZ/32003,  {{0, -9, 4, -2, -4, -9, -10, 6, -8, 0}, 
            {1, 0, -10, 9, 3, -4, 1, 1, -10, -3}, 
	    {5, 7, -4, -5, -7, 7, 4, 6, -3, 2}, 
	    {2, 8, 6, -6, 4, 3, 8, -10, 7, 8}, 
	    {-9, -9, 0, 4, -3, 9, 4, 4, -4, -4}})
phi = affineMakeRingMaps(M,R)
apply (gens(R),r->phi#2 r)
assert ( {4, -10, -4, 6, 0} == apply (gens(R),r->phi#2 r) )

J = affinePointsByIntersection(M,R);
///

TEST ///
R = ZZ/32003[vars(0..4), MonomialOrder=>Lex]
M = matrix(ZZ/32003,  {{0, -9, 4, -2, -4, -9, -10, 6, -8, 0}, 
            {1, 0, -10, 9, 3, -4, 1, 1, -10, -3}, 
	    {5, 7, -4, -5, -7, 7, 4, 6, -3, 2}, 
	    {2, 8, 6, -6, 4, 3, 8, -10, 7, 8}, 
	    {-9, -9, 0, 4, -3, 9, 4, 4, -4, -4}})
phi = affineMakeRingMaps(M,R)
apply (gens(R),r->phi#2 r)
assert ( {4, -10, -4, 6, 0} == apply (gens(R),r->phi#2 r) )

J = affinePointsByIntersection(M,R);
C = affinePoints(M,R);
assert ( J == C_2 )
assert ( C_1 == ideal(e^6,d*e^3,d^2*e,d^3,c,b,a) )
assert ( C_0 == sort apply (standardPairs monomialIdeal C_2, p -> p#0) )
assert (
     (affinePointsMat(M,R))#0 == 
      matrix(ZZ/32003, {{1, -9, 81, -729, 6561, 4957, 2, -18, 162, 4}, {1, -9, 81, -729, 6561,
      4957, 8, -72, 648, 64}, {1, 0, 0, 0, 0, 0, 6, 0, 0, 36}, {1, 4, 16, 64, 256, 1024,
      -6, -24, -96, 36}, {1, -3, 9, -27, 81, -243, 4, -12, 36, 16}, {1, 9, 81, 729, 6561,
      -4957, 3, 27, 243, 9}, {1, 4, 16, 64, 256, 1024, 8, 32, 128, 64}, {1, 4, 16, 64,
      256, 1024, -10, -40, -160, 100}, {1, -4, 16, -64, 256, -1024, 7, -28, 112, 49}, {1,
      -4, 16, -64, 256, -1024, 8, -32, 128, 64}})
)
assert ( first entries transpose (affinePointsMat(M,R))#1 == C_0 )
///

---------------------------------------------------------------------
-- FG: tests for projective and fat points (v3)
---------------------------------------------------------------------

TEST///
     M = id_(ZZ^3)
     R = QQ[x,y,z]
     mults = {2,2,2}
     (Q,inG,G) = affineFatPoints(M,mults,R)
     assert( G == {x^2+2*x*y+y^2+2*x*z+2*y*z+z^2-2*x-2*y-2*z+1,
     x*z^2+y*z^2+z^3-x*z-y*z-2*z^2+z, y^2*z+y*z^2-y*z, x*y*z,
     x*y^2+y^3-y*z^2-x*y-2*y^2+y, z^4-2*z^3+z^2, y*z^3-y*z^2,
     y^4-2*y^3+y^2})
     assert(G == affineFatPointsByIntersection(M,mults,R))
///

TEST///
     M = matrix {{1, 0, 0, 1}, {0, 1, 0, 1}, {0, 0, 1, 1}}
     R = QQ[x,y,z]
     (inG,G) = projectivePoints(M,R)
     assert( G == {x*z-y*z, x*y-y*z, y^2*z-y*z^2})
     assert(G == projectivePointsByIntersection(M,R))
///

TEST///
     M = matrix {{1, 0, 0, 1}, {0, 1, 0, 1}, {0, 0, 1, 1}}
     mults = {1,2,3,4}
     R = QQ[x,y,z]
     (inG,G) = projectiveFatPoints(M,mults,R)
     assert( G == {x^4*z-3*x^3*y*z+3*x^2*y^2*z-x*y^3*z-x^3*z^2+3*x^2*y*z^2-3*x
      *y^2*z^2+y^3*z^2, x^4*y-2*x^3*y^2+x^2*y^3-2*x^3*y*z+4*x^2*y^
      2*z-2*x*y^3*z+x^2*y*z^2-2*x*y^2*z^2+y^3*z^2,
      x^3*y*z^2-3*x^2*y^2*z^2+3*x*y^3*z^2-y^4*z^2-x^3*z^3+3*x^2*y*
      z^3-3*x*y^2*z^3+y^3*z^3,
      x^3*y^2*z-2*x^2*y^3*z+x*y^4*z-2*x^2*y^2*z^2+4*x*y^3*z^2-2*y^
      4*z^2-x^3*z^3+4*x^2*y*z^3-5*x*y^2*z^3+2*y^3*z^3,
      x^3*y^3-x^2*y^4-3*x^2*y^3*z+3*x*y^4*z+3*x*y^3*z^2-3*y^4*z^2-
      x^3*z^3+4*x^2*y*z^3-6*x*y^2*z^3+3*y^3*z^3,
      x^2*y^3*z^2-2*x*y^4*z^2+y^5*z^2-2*x^2*y^2*z^3+4*x*y^3*z^3-2*
      y^4*z^3+x^2*y*z^4-2*x*y^2*z^4+y^3*z^4,
      x^2*y^4*z-x*y^5*z-3*x*y^4*z^2+3*y^5*z^2-3*x^2*y^2*z^3+9*x*y^
      3*z^3-6*y^4*z^3+2*x^2*y*z^4-5*x*y^2*z^4+3*y^3*z^4,
      x^2*y^5-4*x*y^5*z+6*y^5*z^2-4*x^2*y^2*z^3+12*x*y^3*z^3-12*y^
      4*z^3+3*x^2*y*z^4-8*x*y^2*z^4+6*y^3*z^4,
      x*y^5*z^2-y^6*z^2-3*x*y^4*z^3+3*y^5*z^3+3*x*y^3*z^4-3*y^4*z^
      4-x*y^2*z^5+y^3*z^5,
      x*y^6*z-4*y^6*z^2-6*x*y^4*z^3+12*y^5*z^3+8*x*y^3*z^4-12*y^4*
      z^4-3*x*y^2*z^5+4*y^3*z^5,
      y^7*z^2-4*y^6*z^3+6*y^5*z^4-4*y^4*z^5+y^3*z^6})
     assert(G == projectiveFatPointsByIntersection(M,mults,R))
///

end

-*
--test of affinePoints
TEST///
C = affinePoints(M,R);
assert ( J == C_2 )
assert ( C_1 == ideal(e^6,d*e^3,d^2*e,d^3,c,b,a) )
assert ( C_0 == sort apply (standardPairs monomialIdeal C_2, p -> p#0) )
assert (
     (affinePointsMat(M,R))#0 == 
      matrix(ZZ/32003, {{1, -9, 81, -729, 6561, 4957, 2, -18, 162, 4}, {1, -9, 81, -729, 6561,
      4957, 8, -72, 648, 64}, {1, 0, 0, 0, 0, 0, 6, 0, 0, 36}, {1, 4, 16, 64, 256, 1024,
      -6, -24, -96, 36}, {1, -3, 9, -27, 81, -243, 4, -12, 36, 16}, {1, 9, 81, 729, 6561,
      -4957, 3, 27, 243, 9}, {1, 4, 16, 64, 256, 1024, 8, 32, 128, 64}, {1, 4, 16, 64,
      256, 1024, -10, -40, -160, 100}, {1, -4, 16, -64, 256, -1024, 7, -28, 112, 49}, {1,
      -4, 16, -64, 256, -1024, 8, -32, 128, 64}})
)
assert ( first entries transpose (affinePointsMat(M,R))#1 == C_0 )
///
*-

end--
uninstallPackage "Points"
restart
installPackage "Points"
viewHelp Points

check "Points"

 
 
----------------
--------------

toString C_1
restart
errorDepth = 0


uninstallPackage "Points"
installPackage "Points"
R = ZZ/32003[vars(0..4), MonomialOrder=>Lex]
M = matrix(ZZ/32003,  {{0, -9, 4, -2, -4, -9, -10, 6, -8, 0}, 
            {1, 0, -10, 9, 3, -4, 1, 1, -10, -3}, 
	    {5, 7, -4, -5, -7, 7, 4, 6, -3, 2}, 
	    {2, 8, 6, -6, 4, 3, 8, -10, 7, 8}, 
	    {-9, -9, 0, 4, -3, 9, 4, 4, -4, -4}})

phi = affineMakeRingMaps(M,R)
apply (gens(R),r->phi#2 r)
assert ( {4, -10, -4, 6, 0} == apply (gens(R),r->phi#2 r) )


phi#2
time J = affinePointsByIntersection(M,R)
transpose matrix{oo}



time C = points(M,R)
transpose gens ideal C_2

M = random(ZZ^3, ZZ^5)
R = QQ[x,y,z]
phi = affineMakeRingMaps(M,R)
apply (gens(R),r->phi#2 r)
phi#2

R = ZZ/32003[vars(0..4), MonomialOrder=>Lex]
M = random(ZZ^5, ZZ^150)

time J = affinePointsByIntersection(M,R);
transpose matrix{oo}

time C = points(M,R);
transpose gens ideal C_2
assert(J == C_2)

R = ZZ/32003[vars(0..4)]

K = ZZ/32003
R = K[vars(0..7), MonomialOrder=>Lex]
R = K[vars(0..7)]
M = random(K^8, K^500)
time C = points(M,R);
time J = affinePointsByIntersection(M,R);
assert(C_2 == J)

K = ZZ/32003
R = K[x_0 .. x_39]
M = random(K^40, K^80)
time C = points(M,R);


getColumnChange oo_0
apply(Fs, f -> f(a*b*c*d))
B = sort basis(0,2,R)
B = sum(flatten entries basis(0,2,R))
B = matrix{reverse terms B}
P = transpose matrix {apply(Fs, f -> f (transpose B))}
B * syz 
transpose oo
 -- column reduction:

P = mutableMatrix P 
H = new MutableHashTable
reduceColumn(P,null,H,0)
reduceColumn(P,null,H,1)
P
reduceColumn(P,null,H,2)
reduceColumn(P,null,H,3)
reduceColumn(P,null,H,4)
reduceColumn(P,null,H,5)
reduceColumn(P,null,H,6)
reduceColumn(P,null,H,7)
reduceColumn(P,null,H,8)
reduceColumn(P,null,H,9)
P
reduceColumn(P,null,H,10)
reduceColumn(P,null,H,11)
reduceColumn(P,null,H,12)
P

M = matrix{{1,2,3,4}}

K = ZZ/32003
M ** K