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-- The construction of the polynomial system of {6,21,4} --
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We give the method constructing the polynomial system of {6,21,4} 
using a computer algebra system Asir.

Notation: 
The brackets including number, e.g. [0], are prompts of Asir.
Capital letters, e.g. X,Y,G, are variables of Asir.
";" represents the command terminator with screen output.
"$" represents the commnad terminator without screen output.
"//" represents comments. You must not input these lines.
"-->" represents outputs on Asir.

Before starting Asir, we must copy the file "delta.rr" to the folder
"bin" in the folder installed Asir.

[0] load("./delta.rr");
// The required functions are loaded by this command.

[14] X=t^6+a2*t^4+a3*t^3+a4*t^2+a5*t+a6$
[15] Y=t^21+b1*t^20+b2*t^19+b3*t^18+b4*t^17+b5*t^16+b6*t^15+b7*t^14+b8*t^13
+b9*t^12+b10*t^11+b11*t^10+b12*t^9+b13*t^8+b14*t^7+b15*t^6+b16*t^5
+b17*t^4+b18*t^3+b19*t^2+b20*t+b21$
[16] G=y^2-x^7+c00+c10*x+c20*x^2+c30*x^3+c40*x^4+c50*x^5$
//We input polynomials. X corresponds to $x$, Y to $y$ and G to $g_2$
//on page 9 in our preprint. 

[17] CL=coeflist(subst(G,x,X,y,Y),t,4)$
//subst(G,x,X,y,Y) substitutes X to x, Y to y for Polynomial G. 
//coeflist(P,t,4) outputs the list of coefficients of terms in polynomial P 
//with t-degree greater than 4.

[18] length(CL);
-->38
//We get the polynomial system CL with 38 polynomials.

[19] PL=eliminate(CL,b1)$
//We get the polynomial system PL eliminated the variable b1 
//using the linear polynomial w.r.t. b1.

//We execute the following commands to eliminate variables c50,c40,c30,c20,c10,
//b2,b3,b4,b5,b6,b7,b8,b9,b10,b11,b13,b14,b15,b16,b17,b19,b20,b21.
 PL=eliminate(PL,c50)$
 PL=eliminate(PL,c40)$
 PL=eliminate(PL,c30)$
 PL=eliminate(PL,c20)$
 PL=eliminate(PL,c10)$
 PL=eliminate(PL,b2)$
 PL=eliminate(PL,b3)$
 PL=eliminate(PL,b4)$
 PL=eliminate(PL,b5)$
 PL=eliminate(PL,b6)$
 PL=eliminate(PL,b7)$
 PL=eliminate(PL,b8)$
 PL=eliminate(PL,b9)$
 PL=eliminate(PL,b10)$
 PL=eliminate(PL,b11)$
 PL=eliminate(PL,b13)$
 PL=eliminate(PL,b14)$
 PL=eliminate(PL,b15)$
 PL=eliminate(PL,b16)$
 PL=eliminate(PL,b17)$
 PL=eliminate(PL,b19)$
 PL=eliminate(PL,b20)$
 PL=eliminate(PL,b21)$
 vars(PL);
-->[a2,a3,a4,a5,a6,b12,b18]
//As a result, we get the polynomial system with 7 variables 
//[a2,a3,a4,a5,a6,b12,b18] and 14 polys.

[44] PL=list_subst_list(PL,[[a2,a],[a3,b],[a4,c],[a5,d],[a6,e],[b12,f],[b18,g]])$
//We replace a2 with a, a3 with b, a4 with c, a5 with d, a6 with e, 
//b12 with f, b18 with g.

[45] C=PL[13]$
//We get the check polynomial C of {6,21,4}, which is the coefficient 
//of term with t-degree=4 in G.

[46] bsave(C,"./6-21-4_chkpoly");
//We save the polynomial C as a binary data at the file "6-21-4_chkpoly".

[47] PL=omit_nl(PL,[13])$
//We omit 13-th element C from the list PL.
//We get the polynomial system of {6,21,4}.

[48] bsave(PL,"./6-21-4_poly_13");
//We save the polynomial system PL as a binary data at the file 
//"6-21-4_poly_13".

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