, We discard now the spurious solutions

, we get two polynomials F a := z 1 + f a (x 1 , x 2 ) e F b := z 1 + f b (x 1 , x 2 ) and a partition of the 56 points in two subsets Z a , Z b of cardinality 28, satisfying the properties stated in the previous section

, The case of F 8 : cyclic configurations, vol.6

, If we draw the tower structure of these points, disregarding the fourth coordinate as a consequence of proposition 8.2.2, we get a, vol.1, p.1

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, LIBRARY: JMBTest.lib A library for Singular which performs JM basis test

, AUTHOR: Michela Ceria, email: michela.ceria@unito.it

S. Also, JMSConst_lib 10 KEYWORDS: J?marked schemes 11 12 OVERVIEW: 13 The library performs the J?marked basis test

, 14 Such a test is performed via the criterion

F. Cioffi and M. Roggero, Flat Families by Strongly

, Stable Ideals and a Generalization of Groebner Bases

, J. Symbolic Comput, vol.46, 1070.

C. Bertone, F. Cioffi, and P. Lella,

M. Roggero, Upgraded methods for the effective 35 computation of marked schemes on a strongly stable ideal, Journal of Symbolic Computation, vol.36, 2012.

, StartOrderingV(list,list) ordering of polynomials as in

, TestJMark(list) tests whether we have a J?marked basis

. Lib-&quot;qhmoduli,

, LIB "monomialideal.lib

. /-*-&quot;, USAGE: mod_init(

, 50 RETURN: struct: jmp 51 EXAMPLE: example mod_init; ("jmp

{. Example,

/. Usage,

, G list, c int 62 RETURN: list: T

{. Example,

, Terns(G2F, vol.1

, Terns, vol.2

, proc VConst(list G

/. Usage,

G. List,

. //print,

, USAGE: Minimus(L)

G. , , vol.174

{. Example,

, 189 ring r=0

, USAGE: Maximus(L)

G. , RETURN: list: V 197 NOTES: it returns the maximal variable generating the ideal L.@ * 198 The input must be an ideal generated by variables. 199 EXAMPLE: example Maximus

{. Example,

, 211 ring r=0

, proc GJmpMins(jmp P, jmp Q)

, USAGE: GJmpMins(P,Q)

. P-jmp and . Jmp, , vol.218

{. Example,

, 258 ring r=0

. Gjmpmins, , p.2

. Gjmpmins, , p.3

, 270 GJmpMins(p1,p1)

/. Usage:-terncompare, (. , B. , C. );-a-list, and . List,

. Terncompare,

M. Proc,

, USAGE: Minimal

. V-list and . List, , vol.333

{. Example,

, 358 ring r=0

. Minofv,

O. Proc,

, USAGE: OrderingV

. V-list and . List,

. M=minofv,

. V=delete,

(. R=orderingv, G. , and R. ). ,

{. Example,

, 400 ring r=0

, OrderingV

, USAGE: StartOrdina

. V-list and . List, RETURN: list: R 420 NOTE: Input Vm,G. This procedure uses OrderingV to get

{. Example,

, 428 ring r=0

. Startorderingv,

, USAGE: moltiplica(L,G)

, G list 447 RETURN: jmp: K 448 NOTE: Input: a 3?ple

{. Example,

, 460 ring r=0

, 475 Multiply(P,G2F)

, proc IdealOfV(list V)

/. , USAGE: IdealOfV(V)

, 499 //print

. //i=std,

{. Example,

, 506 ring r=0

. /-*-&quot;, USAGE: NewWeight(n)

{. Example,

. Newweight,

, proc FinalVm(list V1 , list G1 , r)

, USAGE: FinalVm(V1, G1, r)

, N=list(

. //print,

, 577 def V=imap(r,V1)

/. G=imap,

, 579 //print(V)

, 580 def MM=imap(r,M

. Ll=multiply,

, LL.t=subst(LL.t,t, vol.1

. Ll and . Ll,

, I=I+ideal(UU.h+UU.t

, 612 attrib(I,"isSB, vol.1

, N=list(

, N=list(

{. Example,

, 650 ring r=0

. Finalvm,

, proc ConstructorMain(list G, int c,r)

/. Usage,

, G list, c int 669 RETURN: list: R 670 NOTE: At the end separated by degree. 671 EXAMPLE: example Costruttore

. Vconst,

, 675 //print

. //print, Ordinare")

(. R=finalvm and G. ,

. //print,

{. Example,

, 694 ring r=0

, ConstructorMain(G2F,6,r)

, proc EKCouples(jmp A, jmp B)

, USAGE: CoppiaEK(A,B)

. //print,

, //print(var(j))

. E=var,

, //print(E)

. //print,

, 739 if(Minimus(variables(B.h))>=Maximus(variables(E)))

. //print,

{. Example,

, 762 EKCouples

, proc EKPolys(list G)

, USAGE: PolysEK(G); G list 767 RETURN: list: EK, list: D 768 NOTE: At the end EK polynomials and their degrees 769 EXAMPLE: example PolysEK

. C=ekcouples,

, 792 //print

, C=insert

(. Ek=insert and C. Ek,

{. Example,

, 810 ring r=0

, 824 EKPolys(G2F)

, USAGE: EKPolynomials(EK,G)

, EK list, G list, vol.829

{. Example,

, 854 list EK,D=EKPolys(G2F)

. Ekpolynomials,

. /-*-&quot;, USAGE: TestJMark(G)("Only One Polynomial"); EK,D=EKPolys(G1)

, 879 //print

, //I found EK couples 881 int massimo=Max(D)

, 882 list V1=ConstructorMain(G1,massimo,r)

, Costruttore")

, 886 int minimo=Min(deg(mi.h))

, 887 intvec u=NewWeight(nvars(r)+1)

, 888 list L=ringlist(r)

, NN=list(

, //print(j)

, //print(j)

. //print,

, JJ, vol.934

, 935 def EK=imap(r,EK

, N=list(

, 946 for(j=size(JJJ[i]); j>0

, 976 p=EKPolynomials(EK

. //print,

. I=idealofv,

{. Example,

, 1003 ring r=0

, TestJMark(G2F,r)

A. , JMBConst.lib: a J-marked schemes constructor

&. Id,

, Algebraic Geometry

, AUTHOR: Michela Ceria, email: michela.ceria@unito.it

F. Cioffi and M. Roggero, Flat Families by Strongly

, Stable Ideals and a Generalization of Groebner Bases

F. Bclr]-cristina-bertone, P. Cioffi, and . Lella, J. Symbolic Comput, vol.46, p.27, 1070.

M. Roggero, Upgraded methods for the effective 29 computation of marked schemes on a strongly stable ideal, 30, Journal of Symbolic Computation, vol.31, 2012.

. Jmarkedscheme,

, USAGE: BorelCheck(Borid,r)

, Borid ideal, r ring 45 RETURN: int: d 46 NOTE: Input must be a monomial ideal

{. Example,

, 86 ring r=0

*. Borid=y^2,

. Borelcheck,

, USAGE: ArrangeBorel(Borid)

, Borid ideal 93 RETURN: list: Input 94 NOTE: Input must be a monomial ideal

, It also returns a list containing the size of every sublist generated. 98 EXAMPLE: example ArrangeBorel

{. Example,

, 119 ring r=0

*. Borid=y^2,

, ArrangeBorel(Borid)

/. Usage,

B. List, NumN list 126 RETURN: int: d 127 NOTE: B is the grouped Borel

*. Numn,

{. Example,

, 142 ring r=0

*. Borid=y^2,

(. Numnewvar, NumN); NewTails(ideal NI

, USAGE: NewTails(NI,s); NI ideal, s int 151 RETURN: list: M 152 NOTE: The procedure construct the tails of the required unknown, J?marked, p.369

{. Example,

(. Newtails and . Ni, , vol.3

, USAGE: ArrangeTails(Q); Q list 174 RETURN: list: Q 175 NOTE: Constructs the final list of J?marked polynomials. 176 EXAMPLE: example FormaInput

. //print,

, //print(i)

. //print,

, //print(i)

, Insert empty list for all intermediate degree 196 between the minimum and the maximum

. //print,

. //print,

, //print(i)

{. Example,

, 209 ring r=0, (x,y,z),rp; 210 ideal Borid=y^2 * z,y * z^2,z^3, p.5

, 211 attrib(Borid,"isSB, vol.1

, 212 list B=ArrangeBorel(Borid)

, //Now I must define the NEW RING, putting the c parameters inside. 225 list L=ringlist(r)

, 238 def Borid=imap(r,Borid

, 239 def N=imap(r,N

, 240 def B=imap(r,B

. M=newtails,

, //print(s)

, Pp.h; Pp.t

. /-*-&quot;, USAGE: mod_init(

, 273 RETURN: struct: jmp 274 EXAMPLE: example mod_init; ("jmp

{. Example,

/. Usage,

, G list, c int 285 RETURN: list: T

{. Example,

, 302 ring r=0

, Terns(G2F, vol.1, p.316

, Terns, vol.2

, proc VConst(list G

/. Usage,

G. List,

. //print,

, 344 if(m>size(G))

. //print,

{. Example,

, 379 ring r=0

, USAGE: Minimus(L)

G. , RETURN: list: V 398 NOTES: it returns the minimal variable generating the ideal L

{. Example,

, 412 ring r=0

, USAGE: Maximus(L)

, G list, c int 419 RETURN: list: V

, if(Minimus(variables(P.h/gcdMon(P.h,Q.h)))<Minimus(variables(Q.h/gcdMon(P.h

. //print, Per Indice")

{. Example,

, 481 ring r=0

. Gpolymin, , p.2

. Gpolymin, , p.3

, 493 GPolyMin(p2,p2)

, proc TernComparer(list A

/. Usage:-terncomparer, (. , B. , C. );-a-list, and . List,

. Terncomparer,

, USAGE: Minimal

. V-list and . List, int: R 556 NOTE: Input=list(terns), G. 557 EXAMPLE: example MinimalV, vol.555

{. Example,

, 580 ring r=0

. Minimalv,

, USAGE: Ordinare

. V-list and . List,

. M=minimalv,

. V=delete,

(. R=orderv, G. , and R. ). ,

{. Example,

, 622 ring r=0

, OrderV

, USAGE: StartOrderingV

. V-list and . List, RETURN: list: R 642 NOTE: Input Vm,G. This procedure uses OrderV to get

{. Example,

, 650 ring r=0

. Startorderingv,

, USAGE: MultiplyJmP(L,G)

, G list 669 RETURN: jmp: K 670 NOTE: Input: a 3?ple

{. Example,

, 682 ring r=0

, MultiplyJmP(P,G2F

/. ,

, USAGE: JmpIdeal(V)

/. Return,

/. Example, example JmpIdeal

, 721 //attrib(I,"isSB, vol.1

/. Example,

, // ring r=0

, 739 //r4.t=poly(0)

, USAGE: NewWeight(n); n int 746 RETURN: intvec: u 747 EXAMPLE: example NewWeight

{. Example,

. Newweight,

, USAGE: FinalVm(V1, G1, r)

, N=list(

. //print,

, 800 def V=imap(r,V1)

/. G=imap,

, 802 //print(V)

, 803 def MM=imap(r,M

. //print,

, 820 for(i=1; i<=size(V)

. //print,

, 823 //print(i)

(. and &. Issb, , vol.1

, 831 //print

. Ll=multiplyjmp,

, LL.t=subst(LL.t,t, vol.1

. Ll and . Ll,

, I=I+ideal(UU.h+UU.t

, 842 attrib(I,"isSB, vol.1

. //print,

, N=list(

, N=list(

{. Example,

, 881 ring r=0

. Finalvm, G2F, r); VmConstructor(list G, int c,r)

/. Usage,

G. , RETURN: list: R 901 NOTE: At the end separated by degree. 902 EXAMPLE: example VmConstructor

. Vconst,

, 906 //print

. //print,

. //print, Ordinare")

/. and G. ,

. //print,

{. Example,

, VmConstructor(G2F,6,r); EKCouples(jmp A, jmp B)

, USAGE: CoppiaEK(A,B)

, A list, B list 945 RETURN: list: L 946 NOTE: At the end the monomials involved by EK. 947 EXAMPLE: example EKCouples

. //print,

, //print(var(j))

. E=var,

, //print(E)

. //print,

, 971 if(Minimus(variables(B.h))>=Maximus(variables(E)))

. //print,

{. Example,

, 994 EKCouples

, USAGE: EKPolynomials(G); G list 999 RETURN: list: EK, list: D 1000 NOTE: At the end EK polynomials and their degrees 1001 1002 EXAMPLE: example EKPolynomials

. C=ekcouples,

, 1025 //print

, C=insert

(. Ek=insert and C. Ek,

{. Example,

, 1043 ring r=0

, 1050 jmp r3

, 1051 r3.h=z * y^2

, 1052 r3.t=?x^2 *

, 1053 jmp r4

, 1057 EKPolynomials(G2F

, USAGE: MultEKPolys(G); G list 1062 RETURN: list: p 1063 NOTE: At the end I obtain the EK polynomials and 1064 their degrees. 1065 EXAMPLE: example MultEKPolys

, 1069 //print

, 1070 jmp q

, 1090 list EK,D=EKPolynomials(G2F

. Multekpolys,

W. , E. K. , D. , and Q. ,

. W-list and . List,

, 1107 int minimo=deg(mini.h); u=NewWeight(nvars(r)+1)

, 1114 list L=ringlist(r)

, 1116 //print(L[2])

, 1117 list ordlist="a

, 1119 def H=ring(L)

, N=list(

M. =n,

, N=list(

, 1160 //print

, 1161 def EK=imap(r,EK

, 1162 def MM=imap(r,M

, 1163 def OO=imap(r,O)

, 1164 def pd=imap(r,pd)

, 1165 list G=list(

, 1166 list N=list(

, 1167 for(i=1; i<=size(MM); i++) 1168 { 1169 for

G. =n,

. //print,

, //print(j)

, 1203 p=MultEKPolys(EK

, M=list(

, 1210 for(i=1; i<= size(V[D[j]?minimo+1]); i++) 1211 { 1212 g=V

, 1213 g.h=(g.h) * t

, 1217 attrib(I,"isSB, vol.1

, 1218 //print(I)

, //print(I)

, 1221 q=reduce(t * p,I)

, 1222 q=subst(q,t,1)

, C=coef(q,pd)

, 1225 for(k=1;k<=size(COEFF);k++)

{. Jms=insert,

, 1232 def Jms=imap

, Jms, p.1233

{. Example,

*. Borid=y^2,

, 1239 attrib(Borid,"isSB, vol.1

, 1240 list B=ArrangeBorel(Borid); (i=1;i<=size(B);i++)

, 1250 } 1251 int qc=NumNewVar

, 1269 def Borid=imap(r,Borid

, 1270 def N=imap(r,N

, 1271 def B=imap(r,B

, 1281 for(j=1;j<=size(B[i]);j++)

. M=newtails,

, 1287 s=s+M, vol.2

, 1288 //print(s); (Q)

, 1292 list EK,D= EKPolynomials(P)

, 1293 int massimo=Max(D)

, 1294 //list V=VConst(P, massimo)

, 1296 list V=VmConstructor(P,massimo,r)

V. , P. , and K. ). , 1297 list W=FinalVm

. //print, I V ridotti in ordine sono

, 1299 //print(W)

W. , E. K. , D. , P. , and K. Jmarkedscheme, 1300 list Jms=SchemeEq

, USAGE: JMarkedScheme(Borid, r)

, Borid ideal, r ring 1306 RETURN: list: Jms 1307 NOTE

. //print, Input is OK")

, 1319 attrib(Borid,"isSB, vol.1

, 1320 list B=ArrangeBorel(Borid); (i=1;i<=size(B);i++)

, 1330 } 1331 int qc=NumNewVar

, 1353 def Borid=imap(r,Borid

, 1354 def N=imap(r,N

, 1355 def B=imap(r,B

, 1365 for(j=1;j<=size(B[i]);j++)

. M=newtails,

, 1371 s=s+M, vol.2

, 1372 //print(s); (Q)

, 1376 list EK,D= EKPolynomials(P)

, 1377 int massimo=Max(D)

, 1378 //list V=VConst(P, massimo)

, 1380 list V=VmConstructor(P,massimo,r)

V. , P. , and K. ). , 1381 list W=FinalVm

. //print, I V ridotti in ordine sono

, 1383 //print(W)

(. Jms=schemeeq, D. Ek, P. , and K. ). , WRONG IDEAL IN INPUT

, 1392 print, It is NOT BOREL

{. Example,

, 1398 ring r=0

*. Borid=y^2,

. Jmarkedscheme,

, 58) 4: (?, ?), (?, ?)

, ? , ? ), (? , ? ), vol.8

, ? , ? ), (? , ? ), (? , ? ), vol.9

, 10: (? , ? ), (? , ? ), (? , ? ), (? , ? )

, ? , ? ), (? , ? ), vol.11

, 13: (? , ? ), (? , ? ), (? , ? ), (? , ? )

, 14: (? , ? ), (? , ? ), (? , ), (? , ? )