--========================================================
-- licensed under GPL v2 or any later version
newPackage(
"RationalPoints",
Version => "0.95",
Date => "Aug 21, 2009",
Authors => {
{Name => "Nathaniel Stapleton", Email => "nat.j.stapleton@gmail.com"}
},
Headline => "find the common zeroes of a set of polynomials with coefficients in a finite field",
Keywords => {"Commutative Algebra"},
DebuggingMode => false
)
--==========================================================
--This program takes in an ideal generated by a list of polynomials with coefficients in a finite field.
--It finds all of the common zeroes of the polynomials (over the coefficient field) and returns them.
--It does this by intelligently testing points in affine n-space.
--Stage 1: Preparation: The generators of the inputted ideal are sorted according to minimum index of support
-- As some polynomials do not contain all of the variables some points do not need to be tested.
-- These points are avoided as much as possible. Instructions for avoiding these points and how to
-- add in the necessary points for the successive polynomial are computed and stored in the
-- Preparation stage.
--Stage 2: Evaluation: After preparation the polynomials are iterated through. In each case the data stored
-- from the preparation process is used to generate potential zeroes and the polynomial is evaluated
-- at these points. There are two possible ways this is done, in most cases one variable is left
-- free in the polynomial so that evaluation provides a one variable polynomial which is then
-- factored. However if the polynomials minimal variable has already been tested then the polynomial
-- is just evaluated at the points that have been found to be zeroes of the previous polynomials.
-- In this way at the end of the process only points that are zeroes of all of the polynomials are
-- left behind.
-------------------------------------------------------------
--Variable names for the interested reader:
-- I - the ideal whose generators the program finds the common zeroes of
-- A - the ring I is contained in
-- k - the coefficient field of A
-- Igens - the generators of I that the program works with
-- els - a list of the elements of the field k
-- theinfo - a list containing several pieces of information related to how potential zeroes should be tested
-- zips - a list containing zeroes, by the end it will contain all of them
-- evaluationspace - a list of potential zeroes (although not necessarily whole points of affine n-space)
export{"rationalPoints", "UseGB", "UseMinGens", "SortGens", "LowMem", "Amount"}
--This function shuffles two lists of lists naively.
--e.g. combineLists({{a},{b}},{{0,1},{2}}) == {{a,0,1},{a,2},{b,0,1},{b,2}}
combineLists = (k,l) -> (
return flatten apply(k, i->apply(l, j-> i|j));
);
--This function takes in a finite field and returns a list of the elements of the field. It can be thought of
--as the underlying set.
fieldElements = (k) -> (
J := ideal k;
p := char k;
els := {};
galoisfield := class k === GaloisField;
if galoisfield then (
x := k.PrimitiveElement; --sometimes k_0 is not the primitive element ie. GF 9
e := 1;
b := 0;
els = els|{0};
while b != 1 do (
b = x^e;
e = e+1;
els = els | {b};
);
);
if not galoisfield and char ring J != 0 then (
d := (degree((flatten entries gens J)_0))_0;
a := (gens k)_0;
coeffs := toList ((set toList (0..p-1)) ^** (d));
for i to # coeffs - 1 do (
x := 0;
for j to d-1 do (
x = x+coeffs_i_j*a^j;
);
els = els | {x};
);
);
if not galoisfield and char ring J == 0 then els = toList (0..p-1);
return els;
);
--orderIt orders a list polynomials from highest minimum index of the support to lowest
orderIt = (l,n) -> (
p := {};
for i from 0 to n-1 do (
for j to #l-1 do (
if index max support l_j == n-i-1 then (
p = p|{l_j};
);
);
);
return p;
);
--shuffleSet is called by superPrep, it provides the instructions for how superCombine should work for a
--given polynomial
shuffleSet = (place, a, z) -> (
x := 0;
y := 0;
result := {};
for i from a to z do (
if member(i,place) then (
result = result | {(0,x)};
x = x+1;
)
else (
result = result | {(1,y)};
);
y = y+1;
);
return result;
);
--sortGens is the function called when the SortGens option is set to true. It applies an automorphism of A to
--each element of Igens. The automorphism just permutes the variables of A. The purpose is to cause the fewest
--number of points to be tested. Besides applying the automorphism to Igens it returns the recipe for the inverse
--automorphism which is applied to each of the zeroes at the very end of the computation.
sortGens = (Igens, A) -> (
s := {};
s = sort apply(gens A, i -> {sum apply(Igens, j -> if isSubset({i}, support j) then 1 else 0),i});
perm := apply(s, i -> i#1);
f := map(A,A,perm);
Igensperm := apply(Igens, i -> f i);
l := apply(perm, i -> index i);
return (Igensperm, l);
)
--superPrep returns two pieces of information, the first is the instructions for which variable to test
--and the second is the variables actually showing up in polynomials (certainly we do not want to test
--points for variables that don't even show up)
superPrep = (Igens, z) -> (
sup := apply(Igens, i -> apply(support i, j -> index j)); --maybe call sup supportofpolys
usedvars := set sup_0;
sup2 := {set sup_0};
place := shuffleSet(toList (set sup_0 - {first sup_0}), min sup_0 + 1, z); --apply(toList (set sup_0 - {first sup_0}), j -> j - min sup_0 - 1);
theinfo := {{place,0, min sup_0, # sup_0 - 1}};
--sup2_i contains the support of the ith poly that is relevant when creating an list of points to be evaluated at.
for i from 1 to #sup-1 do (
if min sup_i < min sup_(i-1) then sup2 = sup2|{set sup_i - {first sup_i}} else sup2 = sup2|{set sup_i};
);
for i from 1 to #sup - 1 do ( --perhaps start this at 1? how about doing sup2 in it's own loop first.
if isSubset(sup2_i,usedvars) then ( --should be sup_i, used to be sup2_i
theinfo = theinfo|{{{},1,min sup_i,0}};
)
else if toList (sup2_i*(usedvars)) == {} then ( --should this be intersecting with usedvars? said set sup_i in place of usedvars
place = shuffleSet(sup2_i, min sup_i + 1, z);
theinfo = theinfo|{{place,2,min sup_i,#sup2_i}};
)
else (
if min toList sup_i == min toList usedvars then (
place = shuffleSet(toList sup2_i - usedvars, min sup_i, z);
)
else (
place = shuffleSet(toList sup2_i - usedvars, min sup_i + 1, z);
);
theinfo = theinfo|{{place,0, min sup_i,# (toList sup2_i - usedvars)}};
);
usedvars = usedvars + set sup_i;
);
return (theinfo, usedvars);
);
--superCombine is the function that needs to be fastest. It takes in two lists of lists and shuffles them
--together based on a recipe (contained in the variable theinfosubi).
superCombine = (theinfosubi, zips, els) -> (
scheme := theinfosubi_0;
case := theinfosubi_1;
n := theinfosubi_3;
if case == 1 then return zips;
newstuff := apply(toList (set els)^**n, deepSplice);
if n == 1 and versionTest() then newstuff = apply(newstuff, i -> {i}); --take out in newest version of M2
return flatten apply(newstuff, k -> apply(zips, l -> apply(scheme, (i,j) -> {k,l}#i#j)));
);
newvar := symbol newvar
--zeroesByFactoring takes in a polynomial and an evaluationspace that does not include one of the variables in the
--polynomial so that the evaluation homomorphism takes values in single variable polynomials which are then factored.
zeroesByFactoring = (thepoly, evaluationspace, A, k, els, numdeadvars) -> (
deadvars := apply(numdeadvars, i->0);
B := k[newvar];
zeroes := {};
memfact := memoize factor;
for i to # evaluationspace - 1 do (
ev := map(B,A,deadvars|{newvar}|(evaluationspace#i));
evk := map (k,B,{0});
q := {};
if isConstant ev thepoly then (
if ev thepoly == 0 then (
q = apply(els,j -> {j});
)
else q = {};
);
if not isConstant ev thepoly then (
p := toList memfact ev thepoly;
p = apply(p,j->(toList j)_0);
for j to #p-1 do (
if (degree p#j)_0 == 1 then q = q|{p#j}; --if p_i == 0 ... else ...
);
q = apply(q,j -> { -evk j }); --now I've got the zeroes of thepoly
);
if evaluationspace_0 === {} then zeroes = q;
if evaluationspace_0 =!= {} then zeroes = zeroes | combineLists(q,{evaluationspace#i});
);
return zeroes;
);
--zeroesByEvaluation evaluates a polynomial on the evaluationspace, a list of possible zeroes.
zeroesByEvaluation = (thepoly, evaluationspace, A, k, numdeadvars) -> (
deadvars := apply(numdeadvars, i->0);
return flatten apply(evaluationspace, i -> (if (map(k,A,deadvars|i)) thepoly == 0 then {i} else {}));
);
findPoints = (Igens, theinfo, els, A, k, m) -> (
z := m - index max support Igens_0 - 1;
zips := {apply(z, i -> 0)};
dif := {};
evaluationspace := superCombine(theinfo_0,zips,els);
zips = zeroesByFactoring(Igens_0, evaluationspace, A, k, els, theinfo_0_2);
temporaryzips := {};
for i from 1 to #Igens - 1 do (
thepoly := Igens_i;
case := (theinfo_i)_1;
dif = index max support Igens_(i-1) - index max support thepoly - 2;
extra := apply(toList (0..dif), i -> 0);
zips = apply(zips, j -> extra|j);
evaluationspace = superCombine(theinfo_i, zips, els);
if max support thepoly == max support Igens_(i-1) then ( --shouldn't this use case == 0 or 1 or something
temporaryzips = zeroesByEvaluation(thepoly, evaluationspace, A, k, (theinfo_i)_2)
) else temporaryzips = zeroesByFactoring(thepoly, evaluationspace, A, k, els, (theinfo_i)_2);
-- if case == 2 then combineLists( -- I need to check and see if this is gonna be worth it)
zips = temporaryzips;
);
return zips;
);
lowMemIterate = (thelist, numels) -> (
thelist#0 = thelist#0+1;
if thelist#0 == numels then (
thelist#0 = 0;
flag := 1;
i := 1;
while flag == 1 do (
thelist#i = thelist#i+1;
if thelist#i == numels then thelist#i = 0 else flag = 0;
i = i+1;
)
);
return thelist;
);
lowMemPoints = (Igens, els, A, k) -> (
loop := # els - 1;
stop := #Igens-1;
endofloop := (loop+1)^(# gens A);
i := 0;
evaluationat := new MutableList from apply(#gens A,i -> 0);
evaluationat#0 = -1;
zeroes := {};
nonzeroflag := 0;
while i < endofloop do (
lowMemIterate(evaluationat, loop+1);
for p to stop do (
if (map(k,A,els_(toList evaluationat))) Igens_p != 0 then (
nonzeroflag = 1;
break;
)
);
if nonzeroflag == 0 then zeroes = zeroes|{els_(toList evaluationat)};
i = i+1;
nonzeroflag = 0;
);
return zeroes;
);
--this tests for a bug in versions of M2 at least less than or equal to 1.2
versionTest = () -> (
return (toList (set {1})^**1)_0 === 1;
);
rationalPoints = method(Options => {Verbose => false, UseGB => false, UseMinGens => false, SortGens => false, LowMem => false, Amount => false})
rationalPoints(Ideal) := opts -> Iprime -> (
-- Initializing the variables. We change the ring so that it has the Lex ordering.
Aprime := ring Iprime;
if not isPolynomialRing Aprime then break; --basic sanity check
k := coefficientRing Aprime;
v := gens Aprime;
m := #v;
J := ideal k;
A := k (monoid [gens Aprime,MonomialOrder => Lex]);
f := map(A,Aprime,gens A);
I := f Iprime;
perm := {};
Igens := {};
if dim k != 0 or char k == 0 or numgens J > 1 then break; --basic sanity check 2
els := {};
els = fieldElements(k);
if opts.UseGB then Igens = flatten entries gens gb I
else if opts.UseMinGens then Igens = flatten entries mingens I
else Igens = flatten entries gens I;
if Igens == {0} then Igens = {};
Igens = orderIt(Igens, m);
if opts.SortGens then (Igens, perm) = sortGens(Igens,A);
zips := {{}};
if opts.Verbose then print Igens;
if Igens === {} then (
if opts.Amount then return (#els)^m;
zips = toList (set els)^**m;
if m == 1 and versionTest() then zips = apply(zips,i -> {i}); --take away this line for new versions
zips = apply(zips,i -> toList i);
return zips;
);
if opts.LowMem or class k === GaloisField then {
if opts.Amount then return # lowMemPoints(Igens, els, A, k);
if not opts.Amount then return lowMemPoints(Igens, els, A, k);
};
--note: as soon as polynomials over GF's can be factored the "or" here should be taken away
unusedvars := {};
(theinfo, usedvars) := superPrep(Igens,m-1);
zips = findPoints(Igens, theinfo, els, A, k, m);
unusedvars = set (0..m-1) - usedvars;
if opts.Amount then return #zips*(#els)^(#unusedvars);
theinfosubfinal := {};
theinfosubfinal = {shuffleSet(unusedvars,0,m-1),0,0,#unusedvars};
zips = superCombine(theinfosubfinal, zips, els);
if opts.SortGens then zips = apply(zips, i -> i_perm);
return zips;
);
--=================================================
beginDocumentation()
----------------------------------------------------
document {
Key => RationalPoints,
Headline => "Find the rational points of an affine variety defined over a finite field",
EM "RationalPoints", " is a package for computing the rational points of an affine variety."
}
document {
Key => {rationalPoints,
(rationalPoints, Ideal),
UseGB,
UseMinGens,
SortGens,
LowMem,
Amount,
[rationalPoints,UseGB],
[rationalPoints,UseMinGens],
[rationalPoints,SortGens],
[rationalPoints,LowMem],
[rationalPoints,Amount],
[rationalPoints,Verbose]
},
Headline => "Compute all of the rational points of an affine variety",
Usage => "l = rationalPoints I",
Inputs => {
"I" => null => {"which is ", ofClass Ideal, " contained in a polynomial ring over a finite field."},
UseGB => Boolean => " turns on and off a Groebner basis computation of the ideal. Default is false.",
UseMinGens => Boolean => " turns on and off a mingens computation of the ideal that may change the chosen generators.",
SortGens => Boolean => " sorts generators in order to make searching for zeroes more efficient.",
LowMem => Boolean => " uses an alternative algorithm that is slower but much less memory intensive.",
Amount => Boolean => " output changes to the number of zeroes.",
Verbose => Boolean => "output includes the generators of the ideal that the computation uses. These may be modified by UseGB or UseMinGens",
},
Outputs => {
"l" => List => { "a list of lists. Each internal list is an n-tuple of elements of the finite field such that the n-tuple represents a point
in Affine n-space lying on the variety defined by the input ideal ", TT "I", ".",},
},
EXAMPLE lines ///
R = ZZ/5[x_1..x_4];
I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);
p = rationalPoints I
///,
PARA {
"This symbol is provided by the package ", TO RationalPoints, "."
}
}
TEST ///
R = ZZ/13[e];
I = ideal(e-2);
assert(# rationalPoints I == 1)
R = ZZ/5[x_1..x_4];
I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);
assert(# rationalPoints I == 8)
k = GF 9;
R = k[m,n,j];
I = ideal(m+a, n*j);
assert(rationalPoints(I,Amount => true) == 17)
R = ZZ/13[b,c,d];
I = ideal (b-b);
assert(rationalPoints(I,Amount => true) == 2197)
assert(#rationalPoints Grassmannian(1,3,CoefficientRing=>ZZ/2) == 36);
///
endPackage "RationalPoints"
--=========================================================================--