This is Maple code corresponding to the paper "A small frame and a certificate of its injectivity" (arXiv:1502.04656)

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Here an injective 4x11 frame V:

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The 4 x4 Hermitian matrix Q, parametrized by the 16 variables x[j, k], y[j, k]:

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The vectors V define frame measurements on Q, which are linear forms in the variables x[j,k], y[j,k]. These are collected into L:

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The matrix Q has rank at most two if all its 3x3 minors vanish. We collect them into the list rk2 and join them with the linear forms L:

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Then we compute a Groebner basis of this collection of polynomials, choosing the monomial order so as to eliminate all the variables by x[3,4] and y[3,4]. The first polynomial is the desired polynomial f:

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Using Sturm sequences, we compute the number of real roots of f(x,1) and find that it has none.

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Our last computation checks that the system of equations Eq=0 has no non-zero solutions with y[3,4]=0. We do this by adjoining y[3,4] and X-1 to our equations for each variable X=x[j,k], y[j,k] and computing a Groebner basis. For each variable, we find a Groebner basis [1], certifying that there are no such solutions.

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(3) |

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