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Merge remote-tracking branch 'upstream/pull/393'
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@DanGrayson
DanGrayson committed Apr 7, 2016
2 parents 418aca0 + 5e91108 commit da9725c
Showing 1 changed file with 48 additions and 32 deletions.
80 changes: 48 additions & 32 deletions M2/Macaulay2/packages/MultipolynomialResultants.m2
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@@ -1,7 +1,7 @@

newPackage(
"MultipolynomialResultants",
Version => "0.2",
Version => "0.3",
Date => "April 6, 2016",
Authors => {{Name => "Giovanni Stagliano'",
Email => "giovannistagliano@gmail.com"
Expand Down Expand Up @@ -45,7 +45,7 @@ Resultant Matrix := (F) -> (
return R;
);
if dim ideal F' > 0 then return sub(0,K);
if min degrees ideal F' == max degrees ideal F' then return Resultant SLtransformation F else error("Resultant not implemented yet for this case");
return Resultant wobble F;
);

internalResultant = (F) -> (
Expand All @@ -69,34 +69,27 @@ internalResultant = (F) -> (
mf=pushFwd(map(A,K[],{}),mf);
Res0^(d_n) * sub(det mf,K)
);

SLtransformation = (F) -> (
-- F row matrix of polynomials
n:=numgens source F -1;
K:=coefficientRing ring F;
A:=matrix 0; d:=0; randomness:=1;
while d == 0 do (
A=matrix(for i to n list for j to n list sub(random(-randomness+1,randomness),K));
d=determinant A
);

wobble = (F) -> (
R:=ring F;
K:=coefficientRing R;
n:=numgens R-1;
A:=matrix 0; d:=0; while not isUnit d do (A=matrix for i to n list for j to n list sub(random(0,1),K); d=det A);
A=d^(-1) * (matrix A_0) | submatrix'(A,,{0});
Sub:=map(R,R,transpose(A*(transpose vars R)));
if verbose then <<"Applied the matrix: "<<A<<endl;
transpose(sub(A,ring F)*transpose(F))
Sub(F)
);

Discriminant=method(TypicalValue => RingElement);

Discriminant RingElement := (G) -> (
-- p. 426 of [Israel M. Gelfand, Mikhail Kapranov, Andrei Zelevinsky-Discriminants, resultants, and multidimensional determinants (1994)];
-- see also [TAKESHI SAITO - THE DISCRIMINANT AND THE DETERMINANT OF A HYPERSURFACE OF EVEN DIMENSION]
if not (isPolynomialRing ring G and isHomogeneous G) then error("expected a homogeneous polynomial");
DisG:=Resultant transpose jacobian matrix{{G}};
if (not isPolynomialRing ring DisG or DisG==0) then return DisG;
-- n:=numgens ring G-2; d:=degree ideal G; a:=((d-1)^(n+2)-(-1)^(n+2))/d;
-- DisG/(d^a)
g:=gcd flatten entries (coefficients DisG)_1;
DisG=DisG/g;
(denominator DisG)^(-1) * numerator(DisG)
if not (isPolynomialRing ring G and isHomogeneous G) then error("expected a homogeneous polynomial");
n:=numgens ring G;
d:=first degree G;
a:=((d-1)^n - (-1)^n)/d;
resG:=Resultant transpose jacobian matrix{{G}};
try lift(resG/(d^a),ring resG) else resG
);

beginDocumentation()
Expand All @@ -105,7 +98,7 @@ document {
Headline => "package for computation of resultants and discriminants",
EM "MultipolynomialResultants", " is a package to compute resultants and discriminants.",
PARA{},
"Let ",TEX///$F_0,\ldots,F_n$///," be ",TEX///$n+1$///," homogeneous polynomials in ",TEX///$n+1$///," variables ",TEX///$x_0,\ldots,x_n$///,". The resultant ",TEX///$R(F_0,\ldots,F_n)$///," is an irreducible polynomial in the coefficients of ",TEX///$F_0,\ldots,F_n$///," which vanishes whenever ",TEX///$F_0,\ldots,F_n$///," have a common root.
"Let ",TEX///$F_0,\ldots,F_n$///," be ",TEX///$n+1$///," homogeneous polynomials in ",TEX///$n+1$///," variables ",TEX///$x_0,\ldots,x_n$///," over a commutative ring ",TEX///$K$///,". The resultant ",TEX///$R(F_0,\ldots,F_n)$///," is a certain polynomial in the coefficients of ",TEX///$F_0,\ldots,F_n$///,"; when ",TEX///$K$///," is an algebraically closed field, ",TEX///$R(F_0,\ldots,F_n)$///," vanishes if and only if ",TEX///$F_0,\ldots,F_n$///," have a common nontrivial root.
The discriminant of a homogeneous polynomial is defined, up to a scalar factor, as the resultant of its partial derivatives. In this package, the resultant is computed, recursively, through the Poisson Formula.",
PARA{},
"For the general theory, see one of the following:",
Expand All @@ -129,13 +122,20 @@ document {
"time Resultant F",
"R=ZZ/97[t_0..t_3];",
"F=matrix{{random(1,R),random(4,R),random(2,R),random(1,R)}}",
"time Resultant F",
"a=symbol a; x=symbol x; R=(QQ[a_(0,0),a_(0,1),a_(0,2),a_(1,0),a_(1,1),a_(1,2),a_(1,3),a_(1,4),a_(1,5),a_(2,0),a_(2,1),a_(2,2),a_(2,3),a_(2,4),a_(2,5),a_(2,6),a_(2,7),a_(2,8),a_(2,9)])[x_0,x_1,x_2];",
"L = a_(0,0)*x_0+a_(0,1)*x_1+a_(0,2)*x_2",
"Q = a_(1,0)*x_0^2+a_(1,1)*x_0*x_1+a_(1,3)*x_1^2+a_(1,2)*x_0*x_2+a_(1,4)*x_1*x_2+a_(1,5)*x_2^2",
"C = a_(2,0)*x_0^3+a_(2,1)*x_0^2*x_1+a_(2,3)*x_0*x_1^2+a_(2,6)*x_1^3+a_(2,2)*x_0^2*x_2+a_(2,4)*x_0*x_1*x_2+a_(2,7)*x_1^2*x_2+a_(2,5)*x_0*x_2^2+a_(2,8)*x_1*x_2^2+a_(2,9)*x_2^3",
"time Resultant matrix{{L,Q,C}}"
},
"time Resultant F"
},
PARA{},
"Below we compute the general expression of the resultant of three forms ", TEX///$L,Q,C$///," on ",TEX///$\mathbb{P}^2$///,", respectively, linear, quadratic and cubic form.",
EXAMPLE {
"-- universal ring of coefficients
ZZ[u_(1,0,0),u_(0,1,0),u_(0,0,1),u_(2,0,0),u_(1,1,0),u_(0,2,0),u_(1,0,1),u_(0,1,1),u_(0,0,2),u_(3,0,0),u_(2,1,0),u_(1,2,0),u_(0,3,0),u_(2,0,1),u_(1,1,1),u_(0,2,1),u_(1,0,2),u_(0,1,2),u_(0,3,0)]",
"-- ring of P^2
oo[t_0,t_1,t_2]",
"L = u_(1,0,0)*t_0+u_(0,1,0)*t_1+u_(0,0,1)*t_2",
"Q = u_(2,0,0)*t_0^2+u_(1,1,0)*t_0*t_1+u_(0,2,0)*t_1^2+u_(1,0,1)*t_0*t_2+u_(0,1,1)*t_1*t_2+u_(0,0,2)*t_2^2",
"C = u_(3,0,0)*t_0^3+u_(2,1,0)*t_0^2*t_1+u_(1,2,0)*t_0*t_1^2+u_(0,3,0)*t_1^3+u_(2,0,1)*t_0^2*t_2+u_(1,1,1)*t_0*t_1*t_2+u_(0,2,1)*t_1^2*t_2+u_(1,0,2)*t_0*t_2^2+u_(0,1,2)*t_1*t_2^2+u_(0,3,0)*t_2^3",
"time Resultant matrix{{L,Q,C}}",
},
SeeAlso => {resultant}
}
document {
Expand Down Expand Up @@ -167,6 +167,7 @@ document {
SeeAlso => {discriminant}
}


TEST /// -- testing resultant
GenericMatrix = (d,K) -> (
-- row matrix of generic homogeneous polynomials of degrees d_0,d_1,...,d_n in n+1 variables
Expand Down Expand Up @@ -215,6 +216,20 @@ SLtransf = (F) -> (
A=d^(-1) * (matrix A_0) | submatrix'(A,,{0});
transpose(sub(A,ring F)*transpose(F))
);
changeVars = (K,u) -> (
d:=for i to random(1,u) list random(1,u+1);
<<"d="<<toSequence d<<endl;
n:=#d -1; x:=local x; R:=K[x_0..x_n];
F:=matrix({for i to n list sub(random(d_i,ZZ[x_0..x_n]),R)});
<<"F="<<toString F<<endl;
pr:=product for i to n list first degree F_(0,i);
A:=matrix for i to n list for j to n list sub(random(-10,10),K);
<<"det A = "<<toString(det A)<<endl;
Sub:=map(R,R,transpose(A*(transpose vars R)));
F':=Sub(F);
<<"F'="<<toString F'<<endl;
(Resultant F') == (det A)^(product toList d) * (Resultant F)
);
L=true; for i from 1 to 5 do for j from 1 to 5 do L= L and compareResultants(i,j,QQ);
for i from 1 to 5 do for j from 1 to 5 do L= L and compareResultants(i,j,ZZ/33331);
assert(L)
Expand All @@ -234,10 +249,11 @@ F=GenericMatrix((2,2),QQ)
assert(Resultant F == Resultant SLtransf F)
R=QQ[t_0,t_1]
assert(Resultant(matrix{{8*t_0*t_1+2*t_1^2, 6*t_0^2+3*t_0*t_1+6*t_1^2}}) == 2160)
-- Resultant matrix{{8*t_0*t_1^2+2*t_1^3, 6*t_0^2+3*t_0*t_1+6*t_1^2}}
assert(Resultant matrix{{8*t_0*t_1^2+2*t_1^3, 6*t_0^2+3*t_0*t_1+6*t_1^2}} == 12960)
R=QQ[x_0..x_6]; X=transpose vars R;
A=matrix for i to 6 list for j to 6 list random(QQ)
assert(Resultant transpose(A*X) == det A)
assert(changeVars(ZZ,3) and changeVars(QQ,3) and changeVars(ZZ[z],2) and changeVars(ZZ/331,4))
///

TEST /// -- testing discriminant
Expand Down

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