Exotic sacks with a three sided die and \(k\) sided die : Magma

To verify our Maple™ results for this calculation, we wanted to program this calculation in Magma as well.

Outline

The approach we used in Magma is different from the approach we used in Maple™ and the approach we used in Macaulay2.

In Magma, we create a cyclotomic field \(K\) generated by \( \zeta_{\mathrm{lcm}(k,l)}\). We create the polynomials \( \mathbf{d}_k\) and \( \widehat{\mathbf{d}}_l\) using exact algebra in the polynomial ring \(K[x]\), and only coerce the coefficients into \(\mathbb{R}\) once at the very end.

Our functions

Given a list of integers (the combinatorial type \( \mathbf{k} \) of a sack) and an integer specifying the desired degree of precision, the function setupSack creates the cyclotomic field \( K \) of order \( \mathrm{lcm}(\mathbf{k})\), the polynomial ring \( R = K[x]\), and the polynomial ring \( \mathbb{R}[x] \) of the desired precision.

setupSack:=function(sack,precision)
lcm:=LCM(sack);
K<z>:=CyclotomicField(lcm);
R<x>:=PolynomialRing(K);
RR:=RealField(precision);
RRX<X>:=PolynomialRing(RR);
return R,x,K,z,RR,RRX;
end function;

The function PartToPolynomial takes a list of integers \( U \) of length \( \lfloor \frac{n-1}{2} \rfloor \) and produces the list of coefficients of the polynomial \[ \prod_{u \in U} (x^2-2 \cos( \frac{2 \pi i u }{n} ) x+1) \] if \(n \) is odd, times an extra factor of \( (x+1) \) if \(n \) is even.

PartToPolynomial :=function(n,U,x)
f:=1;
t:=RootOfUnity(n);
for i:=1 to #U do 
    f:=f*(x-t^(U[i]))*(x-t^(-U[i]));
end for;
if IsEven(n) then 
    f:=f*(x+1);
end if;
return f;
end function;

The function ToRR takes a polynomial in \(K[x]\) and coerces it into \( \mathbb{R}[x]\).

ToRR:=function(f,RRX)
F:=0;
for i:=0 to Degree(f) do
F:=F + ( CoefficientRing(RRX)!(Coefficient(f,i) ))*(RRX.1)^i;
end for;
return F;
end function;

The function AllCoefficientsNonnegative tests whether all the coefficients of a polynomial are nonnegative. Note that in contrast with our Maple™ and Macaulay2 code, we do not bother specifying a cutoff for when a coefficient that is close to zero should be replaced by zero. Instead, we blindly trust Magma's coercion from cyclotomics to real numbers. Our consciences are clean on this point because we already did this calculation in two other computer algebra systems.

AllCoefficientsNonnegative:=function(f)
L:=Coefficients(f);
M:=[L[i] ge 0 : i in [1..#L]];
N:=[true : i in [1..#M]];
return(M eq N);
end function;

The function AllCoefficientsPositive tests whether all the coefficients of a polynomial are positive.

AllCoefficientsPositive:=function(f)
L:=Coefficients(f);
M:=[L[i] gt 0 : i in [1..#L]];
N:=[true : i in [1..#M]];
return(M eq N);
end function;

The input and output of our Magma function EESingleSwaps (where EE stands for exotic examples) are highly analogous to that our Macaulay2 function eeSingleSwaps. This function takes two integers \(k\) and \(l\) and returns the list of exotic pairs of orders \(k\) and \(l\) that can be obtained from a fair pair by a single swap. Each triple in the output [i, j, bool] should be interpreted as follows: the factors \( \chi_i \) and \( \chi_j \) can be swapped to produce an exotic pair, and the value bool is 1 if all the coefficients in the resulting exotic pair are positive and 0 if at least one coefficient is zero.

EESingleSwaps:=function(k,l)
R,x,K,z,RR,RRX:=setupSack([k,l],53);
answer:=[];
m1:= Floor((k-1)/2);
m2:= Floor((l-1)/2);
N1:=[i : i in [1..m1]];
N2:=[i : i in [1..m2]];
for i:=1 to #N1 do 
    for j:=1 to #N2 do 
        if  j/l eq i/k then 
            continue;
        end if;
        U:=Remove(N1,i);
        V:=Remove(N2,j);
        p:=PartToPolynomial(k,U,x);
        p:=p*(x-RootOfUnity(l)^j)*(x-RootOfUnity(l)^-j);
        p:=ToRR(p,RRX);
        q:=PartToPolynomial(l,V,x);
        q:=q*(x-RootOfUnity(k)^i)*(x-RootOfUnity(k)^-i);
        q:=ToRR(q,RRX);
        if AllCoefficientsPositive(p) and AllCoefficientsPositive(q) then
            answer:=Append(answer,[i,j,1]);
        elif AllCoefficientsNonnegative(p) and AllCoefficientsNonnegative(q) then 
            answer:=Append(answer,[i,j,0]);
        end if;
    end for;
end for;
return answer;
end function;

The output

Here is the output.

for k:=3 to 25 do
for l:=3 to 25 do
    print(Sprint(k) cat ", " cat Sprint(l) cat ", " cat Sprint(EESingleSwaps(k,l)));
end for;
end for;

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