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Appendix

We can define the distinguished polynomials related to a cubic function analytically: if f(x) is the given function, we first have to create a homogeneous polynomial in the binary variable (x, y), by the homogenization procedure according to the recipe

Then the determinant

is, up to a numerical factor, the Hessian of the cubic function and the Jacobian of the pair (f, H) with respect to (x, y)

is, again up to a numerical factor, the Jacobian of the cubic function. Applying this recipe to the cubic from equation (1) in the main text gives the numerical factors:

The regular Hessian and Jacobian, as used in the text are then obtained by a process inverse to homogenization: H(x) º H(x, 1)/36 and T(x) º T(x, 1)/108. The writing in last equation is intended to offer a mnemonic rule in order to obtain the Jacobian: the last two coefficients are a kind of mirror image of the first two, in that they are obtained from these by switching places of the indices 0 and 3, replacing the index 1 by 2 and changing the overall sign.

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