, 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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The big Mother of all Dualities 2: Macaulay Bases, Applicable Algebra in Engineering, Communication and Computing archive, vol.17, pp.409-451, 2006. ,
Efficient decoding of (binary) cyclic codes above the correction capacity of the code using Groebner bases, Proc. of ISIT, p.362, 2003. ,
On formulas for decoding binary cyclic codes, Proc. of ISIT, pp.2646-2650, 2007. ,
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Upgraded methods for the effective computation of marked schemes on a strongly stable ideal, J. Symb. Comput, 2012. ,
, A division algorithm in an affine framework for flat families covering Hilbert schemes
A Borel open cover of the Hilbert scheme, J. Symbolic Comput, vol.53, pp.119-135, 2013. ,
Gröbner Bases: An Algorithmic Method in Polynomial Ideal Theory ,
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The Chen-Reed-Helleseth-Truong decoding algorithm and the Gianni-Kalkbrenner Groebner shape theorem, Appl. Algebra Engrg. Comm. Comput, vol.13, issue.3, pp.209-232, 2002. ,
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A library for Singular which performs JM basis test, 2012. ,
A library for Singular which constructs J-Marked Schemes, 2012. ,
, Term-ordering free involutive bases
URL : https://hal.archives-ouvertes.fr/hal-01022881
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Use of Groebner bases to decode binary cyclic codes up to the true minimum distance, IEEE Trans. on Inf. Th, vol.40, issue.5, pp.1654-1661, 1994. ,
Algebraic decoding of cyclic codes: a polynomial ideal point of view, Contemp. Math, vol.168, pp.15-22, 1994. ,
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, SINGULAR 3-1-4 -A computer algebra system for polynomial computations, 2012.
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Computing Groebner Bases for Vanishing Ideals of Finite Sets of Points, 16th International Symposium, AAECC-16, 2004. ,
Efficient computation of zero-dimensional Gröbner bases by change of ordering, J. Symb. Comp, vol.16, pp.329-344, 1993. ,
Rónyai The Lex Game and some applications, J. Symbolic Computation, vol.41, pp.663-681, 2006. ,
Théoreme de division et stabilité en géométrie analytique locale, Ann. Inst. Fourier (Grenoble), vol.29, issue.2, pp.107-184, 1979. ,
Groebner basis structure of finite sets of points, preprint ,
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Involutive bases of Polynomial Ideals, Math. Comp. Simul, vol.45, pp.543-560, 1998. ,
Minimal involutive bases, Math. Comp. Simul, vol.45, pp.519-541, 1998. ,
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, Handbook of Combinatorics, vol.1, 1995.
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Sur la forme canonique des equations algébriques, C.R. Acad. Sci. Paris, vol.157, pp.577-80, 1913. ,
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Standard monomials for q-uniform families and a conjecture of Babai and Frankl, Central European Journal of Mathematics, vol.1, pp.198-207 ,
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Idealistic exponents of singularity In: Algebraic Geometry, The Johns Hopkins Centennial Lectures, pp.52-125, 1977. ,
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, Fundamentals of Error-correcting Codes, 2003.
Sur les systèmes d'équations aux dérivées partelles, J. Math. Pure et Appl, vol.3, pp.65-151, 1920. ,
, Les modules de formes algébriques et la théorie générale des systemes différentiels, Annales scientifiques de l'École Normale Supérieure, 1924.
, Les systèmes d'équations aux dérivées partelles, 1927.
, Lecons sur les systèmes d'équations aux dérivées partelles
, Polynomials that vanish on distinct nth roots of unity. Combinatorics, Probability and Computing, vol.13, pp.37-59
Solving Systems of Algebraic Equations by Using Groebner Bases, L. N. Comp. Sci, vol.378, pp.282-292, 1987. ,
On the stability of Gröbner Bases under specialization, J. Symb. Comp, vol.24, pp.51-58, 1997. ,
, The art of computer programming, vol.3
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The vanishing ideal of a finite set of closed points in affine space, Journal of Pure and Appliead Algebra, vol.212, pp.1116-1133, 2008. ,
Rational components of Hilbert schemes, Rendiconti del Seminario Matematico dell'Università di Padova ,
, di Encyclopedia of Mathematics and its Applications, vol.20, 1997.
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A remark on a remark by Macaulay or Enhancing Lazard Structural Theorem, Bulletin of the Iranian Mathematical Society, vol.29, issue.1, pp.1-45, 2003. ,
, Some Comments on Cerlienco-Mureddu Algorithm and Enhanced Lazard Structural Theorem, 2004.
,
Groebner Bases of Ideals Defined by Functionals with an Application tp Ideals of Projective Points, Applicable Algebra in Engineering, Communication and Computing, vol.4, 1993. ,
The Axis-of-Evil theorem, 2008. ,
Ramella Borel Ideals in three variables, Beiträge zur Algebra und Geometrie, Contributions to Algebra and Geometry, vol.47, issue.1, pp.437-446, 2006. ,
De Nugis Groebnerialium 2: Applying Macaulayâ??s Trick in Order to Easily Write a Groebner Basis, Applicable Algebra in Engineering, Communication and Computing archive, vol.13, pp.409-451, 2003. ,
, Groebner Bases, Coding, and Cryptography, 2009.
General error locator polynomials for binary cyclic codes with t ? 2 and n < 63, BCRI preprint, 2006. ,
Solving polynomial equation systems: Macaulay's paradigm and Groebner technology, 2005. ,
, Wang Partitions and compositions over finite fields, 2012.
Strong Gröbner bases for polynomials over a principal ideal ring, Bull. Austral. Math. Soc, vol.64, pp.505-528, 2001. ,
Correcting errors and erasures via the syndrome variety, J. Pure Appl. Algebra, vol.200, pp.191-226, 2005. ,
General error locator polynomials for binary cyclic codes with t ? 2 and n < 63, IEEE Trans. on Inf. Th, vol.53, pp.1095-1107, 2007. ,
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Sur une questione fondamentale du Calcul intégral Acta mathematica, vol.23, p.203, 1899. ,
, Les systèmes d'équations aux dérivées partielles, 1910.
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Involution -The Formal Theory of Differential Equations and its Applications in Computer Algebra, Algorithms and Computation in Mathematics, vol.24, 2010. ,
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, Coding Theory, 1973.
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, LIBRARY: JMBTest.lib A library for Singular which performs JM basis test
, AUTHOR: Michela Ceria, email: michela.ceria@unito.it
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
Flat Families by Strongly ,
, Stable Ideals and a Generalization of Groebner Bases
, J. Symbolic Comput, vol.46, 1070.
,
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 "monomialideal.lib
, USAGE: mod_init(
, 50 RETURN: struct: jmp 51 EXAMPLE: example mod_init; ("jmp
,
,
, G list, c int 62 RETURN: list: T
,
, Terns(G2F, vol.1
, Terns, vol.2
, proc VConst(list G
,
,
,
, USAGE: Minimus(L)
, , vol.174
,
, 189 ring r=0
, USAGE: Maximus(L)
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 ,
,
, 211 ring r=0
, proc GJmpMins(jmp P, jmp Q)
, USAGE: GJmpMins(P,Q)
, , vol.218
,
, 258 ring r=0
, , p.2
, , p.3
, 270 GJmpMins(p1,p1)
,
,
,
, USAGE: Minimal
, , vol.333
,
, 358 ring r=0
,
,
, USAGE: OrderingV
,
,
,
,
,
, 400 ring r=0
, OrderingV
, USAGE: StartOrdina
RETURN: list: R 420 NOTE: Input Vm,G. This procedure uses OrderingV to get ,
,
, 428 ring r=0
,
, USAGE: moltiplica(L,G)
, G list 447 RETURN: jmp: K 448 NOTE: Input: a 3?ple
,
, 460 ring r=0
, 475 Multiply(P,G2F)
, proc IdealOfV(list V)
USAGE: IdealOfV(V) ,
, 499 //print
,
,
, 506 ring r=0
USAGE: NewWeight(n) ,
,
,
, proc FinalVm(list V1 , list G1 , r)
, USAGE: FinalVm(V1, G1, r)
, N=list(
,
, 577 def V=imap(r,V1)
,
, 579 //print(V)
, 580 def MM=imap(r,M
,
, LL.t=subst(LL.t,t, vol.1
,
, I=I+ideal(UU.h+UU.t
, 612 attrib(I,"isSB, vol.1
, N=list(
, N=list(
,
, 650 ring r=0
,
, proc ConstructorMain(list G, int c,r)
,
, G list, c int 669 RETURN: list: R 670 NOTE: At the end separated by degree. 671 EXAMPLE: example Costruttore
,
, 675 //print
Ordinare") ,
,
,
,
, 694 ring r=0
, ConstructorMain(G2F,6,r)
, proc EKCouples(jmp A, jmp B)
, USAGE: CoppiaEK(A,B)
,
, //print(var(j))
,
, //print(E)
,
, 739 if(Minimus(variables(B.h))>=Maximus(variables(E)))
,
,
, 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
,
, 792 //print
, C=insert
,
,
, 810 ring r=0
, 824 EKPolys(G2F)
, USAGE: EKPolynomials(EK,G)
, EK list, G list, vol.829
,
, 854 list EK,D=EKPolys(G2F)
,
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)
,
, JJ, vol.934
, 935 def EK=imap(r,EK
, N=list(
, 946 for(j=size(JJJ[i]); j>0
, 976 p=EKPolynomials(EK
,
,
,
, 1003 ring r=0
, TestJMark(G2F,r)
JMBConst.lib: a J-marked schemes constructor ,
,
, Algebraic Geometry
, AUTHOR: Michela Ceria, email: michela.ceria@unito.it
Flat Families by Strongly ,
, Stable Ideals and a Generalization of Groebner Bases
, J. Symbolic Comput, vol.46, p.27, 1070.
Upgraded methods for the effective 29 computation of marked schemes on a strongly stable ideal, 30, Journal of Symbolic Computation, vol.31, 2012. ,
,
, USAGE: BorelCheck(Borid,r)
, Borid ideal, r ring 45 RETURN: int: d 46 NOTE: Input must be a monomial ideal
,
, 86 ring r=0
,
,
, 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
,
, 119 ring r=0
,
, ArrangeBorel(Borid)
,
, NumN list 126 RETURN: int: d 127 NOTE: B is the grouped Borel
,
,
, 142 ring r=0
,
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
,
, , 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(i)
,
, //print(i)
, Insert empty list for all intermediate degree 196 between the minimum and the maximum
,
,
, //print(i)
,
, 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
,
, //print(s)
, Pp.h; Pp.t
, USAGE: mod_init(
, 273 RETURN: struct: jmp 274 EXAMPLE: example mod_init; ("jmp
,
,
, G list, c int 285 RETURN: list: T
,
, 302 ring r=0
, Terns(G2F, vol.1, p.316
, Terns, vol.2
, proc VConst(list G
,
,
,
, 344 if(m>size(G))
,
,
, 379 ring r=0
, USAGE: Minimus(L)
RETURN: list: V 398 NOTES: it returns the minimal variable generating the ideal L ,
,
, 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
Per Indice") ,
,
, 481 ring r=0
, , p.2
, , p.3
, 493 GPolyMin(p2,p2)
, proc TernComparer(list A
,
,
, USAGE: Minimal
, int: R 556 NOTE: Input=list(terns), G. 557 EXAMPLE: example MinimalV, vol.555
,
, 580 ring r=0
,
, USAGE: Ordinare
,
,
,
,
,
, 622 ring r=0
, OrderV
, USAGE: StartOrderingV
RETURN: list: R 642 NOTE: Input Vm,G. This procedure uses OrderV to get ,
,
, 650 ring r=0
,
, USAGE: MultiplyJmP(L,G)
, G list 669 RETURN: jmp: K 670 NOTE: Input: a 3?ple
,
, 682 ring r=0
, MultiplyJmP(P,G2F
,
, USAGE: JmpIdeal(V)
,
, example JmpIdeal
, 721 //attrib(I,"isSB, vol.1
,
, // ring r=0
, 739 //r4.t=poly(0)
, USAGE: NewWeight(n); n int 746 RETURN: intvec: u 747 EXAMPLE: example NewWeight
,
,
, USAGE: FinalVm(V1, G1, r)
, N=list(
,
, 800 def V=imap(r,V1)
,
, 802 //print(V)
, 803 def MM=imap(r,M
,
, 820 for(i=1; i<=size(V)
,
, 823 //print(i)
, , vol.1
, 831 //print
,
, LL.t=subst(LL.t,t, vol.1
,
, I=I+ideal(UU.h+UU.t
, 842 attrib(I,"isSB, vol.1
,
, N=list(
, N=list(
,
, 881 ring r=0
G2F, r); VmConstructor(list G, int c,r) ,
,
RETURN: list: R 901 NOTE: At the end separated by degree. 902 EXAMPLE: example VmConstructor ,
,
, 906 //print
,
Ordinare") ,
,
,
,
, 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(var(j))
,
, //print(E)
,
, 971 if(Minimus(variables(B.h))>=Maximus(variables(E)))
,
,
, 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
,
, 1025 //print
, C=insert
,
,
, 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
,
,
,
, 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(
,
, 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
,
,
, //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++)
,
, 1232 def Jms=imap
, Jms, p.1233
,
,
, 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++)
,
, 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)
, 1297 list W=FinalVm
I V ridotti in ordine sono ,
, 1299 //print(W)
, 1300 list Jms=SchemeEq
, USAGE: JMarkedScheme(Borid, r)
, Borid ideal, r ring 1306 RETURN: list: Jms 1307 NOTE
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++)
,
, 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)
, 1381 list W=FinalVm
I V ridotti in ordine sono ,
, 1383 //print(W)
, WRONG IDEAL IN INPUT
, 1392 print, It is NOT BOREL
,
, 1398 ring r=0
,
,
, 58) 4: (?, ?), (?, ?)
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