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MACAULAY2 CODES USED FOR THE PROOFS IN THE PAPER:
"ON LINEAR SYSTEMS OF P3 WITH NINE BASE POINTS" BY M.C.BRAMBILLA, O.DUMITRESCU, E.POSTINGHEL

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CODE FOR P3

--P3
--L_d(m_1,...,m_9), m_i\le 8, m=max(m_i), d\le 2m

KK=ZZ/31991;
E=KK[e_0..e_3];

m=8 --- max multiplicity
d=14 --- degree
N=binomial(d+3,3) --dimension of the space of homogeneous polynomials

f=ideal(e_0..e_3);
fd=f^d;
T=gens gb(fd);
J=jacobian(T);
JJ=jacobian(J);
JJJ=jacobian(JJ);
JJJJ=jacobian(JJJ);
JJJJJ=jacobian(JJJJ);
JJJJJJ=jacobian(JJJJJ);
JJJJJJJ=jacobian(JJJJJJ);

n2=0;n3=0;n4=6;n5=0;n6=0;n7=0;n8=3;
---number of points of multiplicities resp 2,...,8

mat=random(E^1,E^N)*0;
for i from 1 to n2 do (q=random(E^1,E^4),mat=(mat||sub(J,q)));
for i from 1 to n3 do (q=random(E^1,E^4),mat=(mat||sub(JJ,q)));
for i from 1 to n4 do (q=random(E^1,E^4),mat=(mat||sub(JJJ,q)));
for i from 1 to n5 do (q=random(E^1,E^4),mat=(mat||sub(JJJJ,q)));
for i from 1 to n6 do (q=random(E^1,E^4),mat=(mat||sub(JJJJJ,q)));
for i from 1 to n7 do (q=random(E^1,E^4),mat=(mat||sub(JJJJJJ,q)));
for i from 1 to n8 do (q=random(E^1,E^4),mat=(mat||sub(JJJJJJJ,q)));

exprank=n2*binomial(2+2,3)+n3*binomial(2+3,3)+n4*binomial(2+4,3)+n5*binomial(2+5,3)+n6*binomial(2+6,3)+n7*binomial(2+7,3)+n8*binomial(2+8,3);

r=rank mat;

print(r,exprank,(n1,n2,n3,n4,n5,n6,n7,n8), N-r)
-- N-r=affine dimension of the linear system
-- speciality= exprank-r

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CODE FOR P2

-- This program works in P2. Linear systems: L_{3m+2}((m+1)^2,m^8)

KK=ZZ/31991;
E=KK[e_0..e_2];

m=9
d=3*m+2

N=binomial(d+2,2) --dimension of the space of homogeneous polynomials

exprank=8*binomial(m+1,2)+2*binomial(m+3,2)

f=ideal(e_0..e_2);
fd=f^d;
T=gens(fd)
J=jacobian(T)
JJ=jacobian(J)
JJJ=jacobian(JJ);
JJJJ=jacobian(JJJ);
JJJJJ=jacobian(JJJJ);
JJJJJJ=jacobian(JJJJJ);
JJJJJJJ=jacobian(JJJJJJ);
JJJJJJJJ=jacobian(JJJJJJJ);
JJJJJJJJJ=jacobian(JJJJJJJJ);

n2=0;n3=0;n4=0 ;n5=0;n6=0;n7=0;n8=0;n9=8;n10=2;
---number of points of multiplicities resp 2,...,8

mat=random(E^1,E^N)*0;
for i from 1 to n2 do (q=random(E^1,E^3),mat=(mat||sub(J,q)));
for i from 1 to n3 do (q=random(E^1,E^3),mat=(mat||sub(JJ,q)));
for i from 1 to n4 do (q=random(E^1,E^3),mat=(mat||sub(JJJ,q)));
for i from 1 to n5 do (q=random(E^1,E^3),mat=(mat||sub(JJJJ,q)));
for i from 1 to n6 do (q=random(E^1,E^3),mat=(mat||sub(JJJJJ,q)));
for i from 1 to n7 do (q=random(E^1,E^3),mat=(mat||sub(JJJJJJ,q)));
for i from 1 to n8 do (q=random(E^1,E^3),mat=(mat||sub(JJJJJJJ,q)));
for i from 1 to n9 do (q=random(E^1,E^3),mat=(mat||sub(JJJJJJJJ,q)));
for i from 1 to n10 do (q=random(E^1,E^3),mat=(mat||sub(JJJJJJJJJ,q)));

r=rank mat

print(m,exprank,r, N-r)
-- N-r=affine dimension of the linear system
-- speciality= exprank-r

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