#### J. Buhler, Z. Reichstein, *
Symmetric functions and the
phase problem in crystallography, * Transactions of
the American Math. Society, 357 (2005), 2353-2377.

**Abstract:**
This paper was inspired by the work of Herbert
Hauptman, a leading figure in X-ray crystallography and
a 1985 Nobel laureate in chemistry. From a mathematician's
point of view, the situation is roughly as follows.
To determine the structure of a physical
crystal, one needs to know certain quantities, called
``phases". In practice, these cannot be measured
directly; however, one can measure another set
of quantities called ``observables" or
``magnitudes". The question then becomes:
if the ``magnitudes" are known, can one
recover the ``phases", and if so, what is the most efficient
way to carry out the computations? Hauptman and his
collaborators pioneered a probabilistic approach to this
problem. If the number n of atoms in the unit cell
of a crystal is large, these methods work very well.
For smaller n, Hauptman suggested a direct algebraic
approach, i.e., looking for explicit formulas expressing
the magnitudes as rational functions in the observables.
He found such formulas for n <= 3 and asked
if similar formulas exist for other values of n.
We show that the answer is ``yes" by rephrasing
the problem as a question about multiplicative
invariants of a particular finite group action.
Our proofs are constructive, but the resulting
algorithms are, for the most part, too slow
to be of practical significance. For n <= 4,
we develop a faster algorithm, using SAGBI
bases. As a result, we obtain new explicit
formulas for n = 4. We also consider several
related problems and algorithms.

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