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D.15.24.1 nchilb
Procedure from library ncHilb.lib (see ncHilb_lib).
- Usage:
- nchilb(list of relations, an integer, optional);
L is a list of modules (each module represents a free-polynomial),
d is an integer for the degree bound (maximal total degree of
polynomials of the generating set of the input ideal),
#[]=1, computation for non-finitely generated regular ideals,
#[]=2, computation of multi-graded Hilbert series,
#[]=tdeg, for obtaining the truncated Hilbert series up to the
total degree tdeg-1 (tdeg should be > 2), and
#[]=string(p), to print the details about the orbit and system of
equations. Let the orbit is O_I = {T_{w_1}(I),...,T_{w_r}(I)}
($w_i\in W$), where we assume that if T_{w_i}(I)=T_{w_i'}(I)$
for some $w'_i\in W$, then $deg(w_i)\leq deg(w'_i)$.
Then, it prints words description of orbit: w_1,...,w_r.
It also prints the maximal degree and the cardinality of
\sum_j R(w_i, b_j) corresponding to each w_i, where {b_j} is a
basis of I. Moreover, it also prints the linear system (for the
information about adjacency matrix) and its solving time.
- Note :
- A Groebner basis of two-sided ideal of the input should be given in a
special form. This form is a list of modules, where each generator
of every module represents a monomial times a coefficient in the free
associative algebra. The first entry, in each generator, represents a
coefficient and every next entry is a variable.
Ex: module p1=[1,y,z],[-1,z,y], represents the poly y*z-z*y;
module p2=[1,x,z,x],[-1,z,x,z], represents the poly x*z*x-z*x*z
for more details about the input, see examples.
Example:
| | LIB "ncHilb.lib";
ring r=0,(X,Y,Z),dp;
module p1 =[1,Y,Z]; //represents the poly Y*Z
module p2 =[1,Y,Z,X]; //represents the poly Y*Z*X
module p3 =[1,Y,Z,Z,X,Z];
module p4 =[1,Y,Z,Z,Z,X,Z];
module p5 =[1,Y,Z,Z,Z,Z,X,Z];
module p6 =[1,Y,Z,Z,Z,Z,Z,X,Z];
module p7 =[1,Y,Z,Z,Z,Z,Z,Z,X,Z];
module p8 =[1,Y,Z,Z,Z,Z,Z,Z,Z,X,Z];
list l1=list(p1,p2,p3,p4,p5,p6,p7,p8);
nchilb(l1,10);
==>
==> maximal length of words = 2
==>
==> length of the Orbit = 3
==>
==>
==> Hilbert series:
==> 1/(t2-3t+1)
ring r=0,(x,y,z),dp;
==> // ** redefining r (ring r=0,(x,y,z),dp;)
module p1=[1,y,z],[-1,z,y]; //y*z-z*y
module p2=[1,x,z,x],[-1,z,x,z]; // x*z*x-z*x*z
module p3=[1,x,z,z,x,z],[-1,z,x,z,z,x]; // x*z^2*x*z-z*x*z^2*x
module p4=[1,x,z,z,z,x,z];[-1,z,x,z,z,x,x]; // x*z^3*x*z-z*x*z^2*x^2
==> x*gen(7)+x*gen(6)+x*gen(3)+z*gen(5)+z*gen(4)+z*gen(2)-gen(1)
list l2=list(p1,p2,p3,p4);
nchilb(l2,6,1); //third argument '1' is for non-finitely generated case
==>
==> maximal length of words = 3
==>
==> length of the Orbit = 6
==>
==>
==> Hilbert series:
==> 1/(t3+t2-3t+1)
ring r=0,(a,b),dp;
==> // ** redefining r (ring r=0,(a,b),dp;)
module p1=[1,a,a,a];
module p2=[1,a,b,b];
module p3=[1,a,a,b];
list l3=list(p1,p2,p3);
nchilb(l3,5,2);//third argument '2' is to compute multi-graded HS
==>
==> maximal length of words = 3
==>
==> length of the Orbit = 5
==>
==>
==> Hilbert series:
==> (t1^2+t1+1)/(t1*t2^2-t1*t2-t2+1)
ring r=0,(x,y,z),dp;
==> // ** redefining r (ring r=0,(x,y,z),dp;)
module p1=[1,x,z,y,z,x,z];
module p2=[1,x,z,x];
module p3=[1,x,z,y,z,z,x,z];
module p4=[1,y,z];
module p5=[1,x,z,z,x,z];
list l4=list(p1,p2,p3,p4,p5);
nchilb(l4,7,"p"); //third argument "p" is to print the details
==>
==> maximal length of words = 3
==>
==> length of the Orbit = 6
==> words description of the Orbit:
==> 1 x(1) y(1) x(1)*z(2) y(1)*z(2) x(1)*z(2)*z(3)
==>
==> maximal degree, #(sum_j R(w,w_j))
==> NULL
==> 6, 4
==> 1, 1
==> 5, 4
==> 0, 1
==> 2, 1
==>
==> linear system:
==> H(1) = (t)*H(2) + (t)*H(3) + (t)*H(1) + 1
==> H(2) = (t)*H(2) + (t)*H(3) + (t)*H(4) + 1
==> H(3) = (t)*H(2) + (t)*H(3) + (t)*H(5) + 1
==> H(4) = (t)*H(5) + (t)*H(3) + (t)*H(6) + 1
==> H(5) = (t)*H(5) + (t)*H(5) + (t)*H(5) + 0
==> H(6) = (t)*H(3) + (t)*H(3) + (t)*H(1) + 1
==> where H(1) represents the series corresp. to input ideal
==> and i^th summand in the rhs of an eqn. is according
==> to the right colon map corresp. to the i^th variable
==>
==>
==> Hilbert series:
==> (t3+t2+1)/(2t5-2t4-t3+2t2-3t+1)
// of the orbit and system
ring r=0,(x,y,z),dp;
==> // ** redefining r (ring r=0,(x,y,z),dp;)
module p1=[1,y,z,z];
module p2=[1,y,y,z];
module p3=[1,x,z,z];
module p4=[1,x,z,y];
module p5=[1,x,y,z];
module p6=[1,x,y,y];
module p7=[1,x,x,z];
module p8=[1,x,x,y];
module p9=[1,y,z,y,z];
module p10=[1,y,z,x,z];
module p11=[1,y,z,x,y];
module p12=[1,x,z,x,z];
module p13=[1,x,z,x,y];
module p14=[1,x,y,x,z];
module p15=[1,x,y,x,y];
module p16=[1,y,z,y,x,z];
module p17=[1,y,z,y,x,y];
module p18=[1,y,z,y,y,x,z];
module p19=[1,y,z,y,y,x,y];
module p20=[1,y,z,y,y,y,x,z];
module p21=[1,y,z,y,y,y,x,y];
list l5=list(p1,p2,p3,p4,p5,p6,p7,p8,p9,p10,p11,p12,p13,
p14,p15,p16,p17,p18,p19,p20,p21);
nchilb(l5,7,1,2,"p");
==>
==> maximal length of words = 3
==>
==> length of the Orbit = 7
==> words description of the Orbit:
==> 1 x(1) y(1) x(1)*x(2) y(1)*y(2) y(1)*z(2) x(1)*x(2)*y(3\
)
==>
==> maximal degree, #(sum_j R(w,w_j))
==> NULL
==> 3, 10
==> 6, 11
==> 2, 4
==> 2, 2
==> 5, 10
==> 0, 1
==>
==> linear system:
==> H(1) = (t1)*H(2) + (t2)*H(3) + (t3)*H(1) + 1
==> H(2) = (t1)*H(4) + (t2)*H(4) + (t3)*H(4) + 1
==> H(3) = (t1)*H(2) + (t2)*H(5) + (t3)*H(6) + 1
==> H(4) = (t1)*H(4) + (t2)*H(7) + (t3)*H(7) + 1
==> H(5) = (t1)*H(2) + (t2)*H(5) + (t3)*H(7) + 1
==> H(6) = (t1)*H(4) + (t2)*H(6) + (t3)*H(7) + 1
==> H(7) = (t1)*H(7) + (t2)*H(7) + (t3)*H(7) + 0
==> where H(1) represents the series corresp. to input ideal
==>
==>
==> Hilbert series:
==> (-t1*t2-t1*t3-t2*t3-1)/(t1*t2*t3-t1*t2-t1*t3+t1-t2*t3+t2+t3-1)
nchilb(l5,7,1,2,11,"p");
==>
==> Truncation degree = 11
==>
==> maximal length of words = 10
==>
==> length of the Orbit = 60
==> words description of the Orbit:
==> 1 x(1) y(1) z(1) x(1)*x(2) y(1)*x(2) y(1)*y(2) y(1)*\
z(2) z(1)*y(2) z(1)*z(2) x(1)*x(2)*x(3) x(1)*x(2)*y(3) y(1\
)*y(2)*x(3) y(1)*y(2)*y(3) y(1)*z(2)*y(3) z(1)*y(2)*z(3) z(1)\
*z(2)*y(3) z(1)*z(2)*z(3) x(1)*x(2)*x(3)*x(4) y(1)*y(2)*y(3)*x(4\
) y(1)*y(2)*y(3)*y(4) y(1)*z(2)*y(3)*y(4) z(1)*y(2)*z(3)*y(4) \
z(1)*z(2)*y(3)*z(4) z(1)*z(2)*z(3)*y(4) z(1)*z(2)*z(3)*z(4) x(1\
)*x(2)*x(3)*x(4)*x(5) y(1)*y(2)*y(3)*y(4)*x(5) y(1)*y(2)*y(3)*y(4)*\
y(5) y(1)*z(2)*y(3)*y(4)*y(5) z(1)*y(2)*z(3)*y(4)*y(5) z(1)*z(2)\
*y(3)*z(4)*y(5) z(1)*z(2)*z(3)*y(4)*z(5) z(1)*z(2)*z(3)*z(4)*y(5) \
z(1)*z(2)*z(3)*z(4)*z(5) x(1)*x(2)*x(3)*x(4)*x(5)*x(6) y(1)*y(2)*\
y(3)*y(4)*y(5)*x(6) y(1)*y(2)*y(3)*y(4)*y(5)*y(6) z(1)*y(2)*z(3)*y(\
4)*y(5)*y(6) z(1)*z(2)*y(3)*z(4)*y(5)*y(6) z(1)*z(2)*z(3)*y(4)*z(5)\
*y(6) z(1)*z(2)*z(3)*z(4)*z(5)*y(6) z(1)*z(2)*z(3)*z(4)*z(5)*z(6) \
x(1)*x(2)*x(3)*x(4)*x(5)*x(6)*x(7) y(1)*y(2)*y(3)*y(4)*y(5)*y(6)*x(7\
) y(1)*y(2)*y(3)*y(4)*y(5)*y(6)*y(7) z(1)*z(2)*y(3)*z(4)*y(5)*y(6)*\
y(7) z(1)*z(2)*z(3)*y(4)*z(5)*y(6)*y(7) z(1)*z(2)*z(3)*z(4)*z(5)*z(\
6)*y(7) z(1)*z(2)*z(3)*z(4)*z(5)*z(6)*z(7) x(1)*x(2)*x(3)*x(4)*x(5)\
*x(6)*x(7)*x(8) y(1)*y(2)*y(3)*y(4)*y(5)*y(6)*y(7)*x(8) y(1)*y(2)*y\
(3)*y(4)*y(5)*y(6)*y(7)*y(8) z(1)*z(2)*z(3)*y(4)*z(5)*y(6)*y(7)*y(8) \
z(1)*z(2)*z(3)*z(4)*z(5)*z(6)*z(7)*y(8) z(1)*z(2)*z(3)*z(4)*z(5)*z(6\
)*z(7)*z(8) x(1)*x(2)*x(3)*x(4)*x(5)*x(6)*x(7)*x(8)*x(9) y(1)*y(2)*\
y(3)*y(4)*y(5)*y(6)*y(7)*y(8)*x(9) y(1)*y(2)*y(3)*y(4)*y(5)*y(6)*y(7)*\
y(8)*y(9) x(1)*x(2)*x(3)*x(4)*x(5)*x(6)*x(7)*x(8)*x(9)*x(10)
==>
==> maximal degree, #(sum_j R(w,w_j))
==> NULL
==> 3, 10
==> 6, 11
==> NULL
==> 2, 4
==> 3, 10
==> 2, 2
==> 5, 10
==> 6, 11
==> NULL
==> 2, 4
==> 0, 1
==> 3, 10
==> 2, 2
==> 4, 8
==> 5, 10
==> 6, 11
==> NULL
==> 2, 4
==> 3, 10
==> 2, 2
==> 3, 6
==> 4, 8
==> 5, 10
==> 6, 11
==> NULL
==> 2, 4
==> 3, 10
==> 2, 2
==> 2, 4
==> 3, 6
==> 4, 8
==> 5, 10
==> 6, 11
==> NULL
==> 2, 4
==> 3, 10
==> 2, 2
==> 2, 4
==> 3, 6
==> 4, 8
==> 6, 11
==> NULL
==> 2, 4
==> 3, 10
==> 2, 2
==> 2, 4
==> 3, 6
==> 6, 11
==> NULL
==> 2, 4
==> 3, 10
==> 2, 2
==> 2, 4
==> 6, 11
==> NULL
==> 2, 4
==> 3, 10
==> 2, 2
==> 2, 4
==>
==> linear system:
==> H(1) = (t1)*H(2) + (t2)*H(3) + (t3)*H(4) + 1
==> H(2) = (t1)*H(5) + (t2)*H(5) + (t3)*H(5) + 1
==> H(3) = (t1)*H(6) + (t2)*H(7) + (t3)*H(8) + 1
==> H(4) = (t1)*H(6) + (t2)*H(9) + (t3)*H(10) + 1
==> H(5) = (t1)*H(11) + (t2)*H(12) + (t3)*H(12) + 1
==> H(6) = (t1)*H(11) + (t2)*H(11) + (t3)*H(11) + 1
==> H(7) = (t1)*H(13) + (t2)*H(14) + (t3)*H(12) + 1
==> H(8) = (t1)*H(11) + (t2)*H(15) + (t3)*H(12) + 1
==> H(9) = (t1)*H(13) + (t2)*H(14) + (t3)*H(16) + 1
==> H(10) = (t1)*H(13) + (t2)*H(17) + (t3)*H(18) + 1
==> H(11) = (t1)*H(19) + (t2)*H(12) + (t3)*H(12) + 1
==> H(12) = (t1)*H(12) + (t2)*H(12) + (t3)*H(12) + 0
==> H(13) = (t1)*H(19) + (t2)*H(19) + (t3)*H(19) + 1
==> H(14) = (t1)*H(20) + (t2)*H(21) + (t3)*H(12) + 1
==> H(15) = (t1)*H(19) + (t2)*H(22) + (t3)*H(12) + 1
==> H(16) = (t1)*H(19) + (t2)*H(23) + (t3)*H(12) + 1
==> H(17) = (t1)*H(20) + (t2)*H(21) + (t3)*H(24) + 1
==> H(18) = (t1)*H(20) + (t2)*H(25) + (t3)*H(26) + 1
==> H(19) = (t1)*H(27) + (t2)*H(12) + (t3)*H(12) + 1
==> H(20) = (t1)*H(27) + (t2)*H(27) + (t3)*H(27) + 1
==> H(21) = (t1)*H(28) + (t2)*H(29) + (t3)*H(12) + 1
==> H(22) = (t1)*H(27) + (t2)*H(30) + (t3)*H(12) + 1
==> H(23) = (t1)*H(27) + (t2)*H(31) + (t3)*H(12) + 1
==> H(24) = (t1)*H(27) + (t2)*H(32) + (t3)*H(12) + 1
==> H(25) = (t1)*H(28) + (t2)*H(29) + (t3)*H(33) + 1
==> H(26) = (t1)*H(28) + (t2)*H(34) + (t3)*H(35) + 1
==> H(27) = (t1)*H(36) + (t2)*H(12) + (t3)*H(12) + 1
==> H(28) = (t1)*H(36) + (t2)*H(36) + (t3)*H(36) + 1
==> H(29) = (t1)*H(37) + (t2)*H(38) + (t3)*H(12) + 1
==> H(30) = (t1)*H(36) + (t2)*H(38) + (t3)*H(12) + 1
==> H(31) = (t1)*H(36) + (t2)*H(39) + (t3)*H(12) + 1
==> H(32) = (t1)*H(36) + (t2)*H(40) + (t3)*H(12) + 1
==> H(33) = (t1)*H(36) + (t2)*H(41) + (t3)*H(12) + 1
==> H(34) = (t1)*H(37) + (t2)*H(38) + (t3)*H(41) + 1
==> H(35) = (t1)*H(37) + (t2)*H(42) + (t3)*H(43) + 1
==> H(36) = (t1)*H(44) + (t2)*H(12) + (t3)*H(12) + 1
==> H(37) = (t1)*H(44) + (t2)*H(44) + (t3)*H(44) + 1
==> H(38) = (t1)*H(45) + (t2)*H(46) + (t3)*H(12) + 1
==> H(39) = (t1)*H(44) + (t2)*H(46) + (t3)*H(12) + 1
==> H(40) = (t1)*H(44) + (t2)*H(47) + (t3)*H(12) + 1
==> H(41) = (t1)*H(44) + (t2)*H(48) + (t3)*H(12) + 1
==> H(42) = (t1)*H(45) + (t2)*H(46) + (t3)*H(48) + 1
==> H(43) = (t1)*H(45) + (t2)*H(49) + (t3)*H(50) + 1
==> H(44) = (t1)*H(51) + (t2)*H(12) + (t3)*H(12) + 1
==> H(45) = (t1)*H(51) + (t2)*H(51) + (t3)*H(51) + 1
==> H(46) = (t1)*H(52) + (t2)*H(53) + (t3)*H(12) + 1
==> H(47) = (t1)*H(51) + (t2)*H(53) + (t3)*H(12) + 1
==> H(48) = (t1)*H(51) + (t2)*H(54) + (t3)*H(12) + 1
==> H(49) = (t1)*H(52) + (t2)*H(53) + (t3)*H(54) + 1
==> H(50) = (t1)*H(52) + (t2)*H(55) + (t3)*H(56) + 1
==> H(51) = (t1)*H(57) + (t2)*H(12) + (t3)*H(12) + 1
==> H(52) = (t1)*H(57) + (t2)*H(57) + (t3)*H(57) + 1
==> H(53) = (t1)*H(58) + (t2)*H(59) + (t3)*H(12) + 1
==> H(54) = (t1)*H(57) + (t2)*H(59) + (t3)*H(12) + 1
==> H(55) = (t1)*H(58) + (t2)*H(59) + (t3)*H(59) + 1
==> H(56) = (t1)*H(58) + (t2)*H(58) + (t3)*H(58) + 1
==> H(57) = (t1)*H(60) + (t2)*H(12) + (t3)*H(12) + 1
==> H(58) = (t1)*H(60) + (t2)*H(60) + (t3)*H(60) + 1
==> H(59) = (t1)*H(60) + (t2)*H(60) + (t3)*H(12) + 1
==> H(60) = (t1)*H(12) + (t2)*H(12) + (t3)*H(12) + 1
==> where H(1) represents the series corresp. to input ideal
==>
==>
==> Hilbert series:
==> (t1^10+2*t1^9*t2+2*t1^9*t3+t1^9+2*t1^8*t2^2+4*t1^8*t2*t3+2*t1^8*t2+2*t1^8\
*t3^2+2*t1^8*t3+t1^8+2*t1^7*t2^3+4*t1^7*t2^2*t3+2*t1^7*t2^2+4*t1^7*t2*t3^\
2+4*t1^7*t2*t3+2*t1^7*t2+2*t1^7*t3^3+2*t1^7*t3^2+2*t1^7*t3+t1^7+2*t1^6*t2\
^4+4*t1^6*t2^3*t3+2*t1^6*t2^3+4*t1^6*t2^2*t3^2+4*t1^6*t2^2*t3+2*t1^6*t2^2\
+4*t1^6*t2*t3^3+4*t1^6*t2*t3^2+4*t1^6*t2*t3+2*t1^6*t2+2*t1^6*t3^4+2*t1^6*\
t3^3+2*t1^6*t3^2+2*t1^6*t3+t1^6+2*t1^5*t2^5+4*t1^5*t2^4*t3+2*t1^5*t2^4+4*\
t1^5*t2^3*t3^2+4*t1^5*t2^3*t3+2*t1^5*t2^3+4*t1^5*t2^2*t3^3+4*t1^5*t2^2*t3\
^2+4*t1^5*t2^2*t3+2*t1^5*t2^2+4*t1^5*t2*t3^4+4*t1^5*t2*t3^3+4*t1^5*t2*t3^\
2+4*t1^5*t2*t3+2*t1^5*t2+2*t1^5*t3^5+2*t1^5*t3^4+2*t1^5*t3^3+2*t1^5*t3^2+\
2*t1^5*t3+t1^5+2*t1^4*t2^6+4*t1^4*t2^5*t3+2*t1^4*t2^5+4*t1^4*t2^4*t3^2+4*\
t1^4*t2^4*t3+2*t1^4*t2^4+4*t1^4*t2^3*t3^3+4*t1^4*t2^3*t3^2+4*t1^4*t2^3*t3\
+2*t1^4*t2^3+4*t1^4*t2^2*t3^4+4*t1^4*t2^2*t3^3+4*t1^4*t2^2*t3^2+4*t1^4*t2\
^2*t3+2*t1^4*t2^2+4*t1^4*t2*t3^5+4*t1^4*t2*t3^4+4*t1^4*t2*t3^3+4*t1^4*t2*\
t3^2+4*t1^4*t2*t3+2*t1^4*t2+2*t1^4*t3^6+2*t1^4*t3^5+2*t1^4*t3^4+2*t1^4*t3\
^3+2*t1^4*t3^2+2*t1^4*t3+t1^4+2*t1^3*t2^7+5*t1^3*t2^6*t3+2*t1^3*t2^6+5*t1\
^3*t2^5*t3^2+4*t1^3*t2^5*t3+2*t1^3*t2^5+4*t1^3*t2^4*t3^3+4*t1^3*t2^4*t3^2\
+4*t1^3*t2^4*t3+2*t1^3*t2^4+4*t1^3*t2^3*t3^4+4*t1^3*t2^3*t3^3+4*t1^3*t2^3\
*t3^2+4*t1^3*t2^3*t3+2*t1^3*t2^3+4*t1^3*t2^2*t3^5+4*t1^3*t2^2*t3^4+4*t1^3\
*t2^2*t3^3+4*t1^3*t2^2*t3^2+4*t1^3*t2^2*t3+2*t1^3*t2^2+4*t1^3*t2*t3^6+4*t\
1^3*t2*t3^5+4*t1^3*t2*t3^4+4*t1^3*t2*t3^3+4*t1^3*t2*t3^2+4*t1^3*t2*t3+2*t\
1^3*t2+2*t1^3*t3^7+2*t1^3*t3^6+2*t1^3*t3^5+2*t1^3*t3^4+2*t1^3*t3^3+2*t1^3\
*t3^2+2*t1^3*t3+t1^3+2*t1^2*t2^8+5*t1^2*t2^7*t3+2*t1^2*t2^7+6*t1^2*t2^6*t\
3^2+5*t1^2*t2^6*t3+2*t1^2*t2^6+5*t1^2*t2^5*t3^3+5*t1^2*t2^5*t3^2+4*t1^2*t\
2^5*t3+2*t1^2*t2^5+4*t1^2*t2^4*t3^4+4*t1^2*t2^4*t3^3+4*t1^2*t2^4*t3^2+4*t\
1^2*t2^4*t3+2*t1^2*t2^4+4*t1^2*t2^3*t3^5+4*t1^2*t2^3*t3^4+4*t1^2*t2^3*t3^\
3+4*t1^2*t2^3*t3^2+4*t1^2*t2^3*t3+2*t1^2*t2^3+4*t1^2*t2^2*t3^6+4*t1^2*t2^\
2*t3^5+4*t1^2*t2^2*t3^4+4*t1^2*t2^2*t3^3+4*t1^2*t2^2*t3^2+4*t1^2*t2^2*t3+\
2*t1^2*t2^2+4*t1^2*t2*t3^7+4*t1^2*t2*t3^6+4*t1^2*t2*t3^5+4*t1^2*t2*t3^4+4\
*t1^2*t2*t3^3+4*t1^2*t2*t3^2+4*t1^2*t2*t3+2*t1^2*t2+2*t1^2*t3^8+2*t1^2*t3\
^7+2*t1^2*t3^6+2*t1^2*t3^5+2*t1^2*t3^4+2*t1^2*t3^3+2*t1^2*t3^2+2*t1^2*t3+\
t1^2+2*t1*t2^9+5*t1*t2^8*t3+2*t1*t2^8+6*t1*t2^7*t3^2+5*t1*t2^7*t3+2*t1*t2\
^7+6*t1*t2^6*t3^3+6*t1*t2^6*t3^2+5*t1*t2^6*t3+2*t1*t2^6+5*t1*t2^5*t3^4+5*\
t1*t2^5*t3^3+5*t1*t2^5*t3^2+4*t1*t2^5*t3+2*t1*t2^5+4*t1*t2^4*t3^5+4*t1*t2\
^4*t3^4+4*t1*t2^4*t3^3+4*t1*t2^4*t3^2+4*t1*t2^4*t3+2*t1*t2^4+4*t1*t2^3*t3\
^6+4*t1*t2^3*t3^5+4*t1*t2^3*t3^4+4*t1*t2^3*t3^3+4*t1*t2^3*t3^2+4*t1*t2^3*\
t3+2*t1*t2^3+4*t1*t2^2*t3^7+4*t1*t2^2*t3^6+4*t1*t2^2*t3^5+4*t1*t2^2*t3^4+\
4*t1*t2^2*t3^3+4*t1*t2^2*t3^2+4*t1*t2^2*t3+2*t1*t2^2+4*t1*t2*t3^8+4*t1*t2\
*t3^7+4*t1*t2*t3^6+4*t1*t2*t3^5+4*t1*t2*t3^4+4*t1*t2*t3^3+4*t1*t2*t3^2+4*\
t1*t2*t3+2*t1*t2+2*t1*t3^9+2*t1*t3^8+2*t1*t3^7+2*t1*t3^6+2*t1*t3^5+2*t1*t\
3^4+2*t1*t3^3+2*t1*t3^2+2*t1*t3+t1+t2^10+2*t2^9*t3+t2^9+2*t2^8*t3^2+2*t2^\
8*t3+t2^8+2*t2^7*t3^3+2*t2^7*t3^2+2*t2^7*t3+t2^7+2*t2^6*t3^4+2*t2^6*t3^3+\
2*t2^6*t3^2+2*t2^6*t3+t2^6+2*t2^5*t3^5+2*t2^5*t3^4+2*t2^5*t3^3+2*t2^5*t3^\
2+2*t2^5*t3+t2^5+2*t2^4*t3^6+2*t2^4*t3^5+2*t2^4*t3^4+2*t2^4*t3^3+2*t2^4*t\
3^2+2*t2^4*t3+t2^4+2*t2^3*t3^7+2*t2^3*t3^6+2*t2^3*t3^5+2*t2^3*t3^4+2*t2^3\
*t3^3+2*t2^3*t3^2+2*t2^3*t3+t2^3+2*t2^2*t3^8+2*t2^2*t3^7+2*t2^2*t3^6+2*t2\
^2*t3^5+2*t2^2*t3^4+2*t2^2*t3^3+2*t2^2*t3^2+2*t2^2*t3+t2^2+2*t2*t3^9+2*t2\
*t3^8+2*t2*t3^7+2*t2*t3^6+2*t2*t3^5+2*t2*t3^4+2*t2*t3^3+2*t2*t3^2+2*t2*t3\
+t2+t3^10+t3^9+t3^8+t3^7+t3^6+t3^5+t3^4+t3^3+t3^2+t3+1)
//'11' is to compute the truncated HS up to degree 10.
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