I got stuck today on a problem that seems trivial but is in fact not at all trivial: Given N inputs, how to make all possible dimensionless monomials of those N inputs, at (or less than) degree d. For our purposes (which are the units-equivariant machine-learning projects I have with Soledad Villar, JHU), a dimensionless monomial of degree d is a product of integer powers of the inputs, in which those powers can be positive or negative, such that the dimensions cancel out completely, and for which the the max (or sum or some norm) of the absolute values of the powers is ≤d. We have a complete basis of dimensionless monomials, such that any valid dimensionless monomial can be expressed as a monomial of the basis monomials. Because of this, the dimensionless monomials can be seen as the vertices of a Bravais lattice, technically. The problem is just to traverse the entire lattice within some kind of ball. Why is this hard? Or am I just dull? I feel like there are fill algorithms that should do this correctly.
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