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Background

What I think of Ehrhart theory (http://en.wikipedia.org/wiki/Ehrhart_polynomial) asserts that if we take a lattice polytope $P$, and count the number of lattice points in the $t$th dilation of $P$, the result is polynomial in $t$.

If, however, $P$ is only a rational polytope, then in general we get a quasi-polynomial; that is, there is some period $N$, so that if for any $k$, I only look at $t$ congruent to $k$ mod $N$, then again I have a polynomial behavior; however, different residue classes will in general result in different polynomials.

One might expect that any two distinct residue classes have distinct polynomials, but this need not be the case: two different residue classes can have the same polynomial. These are the "accidental" equalities of the question title.

A simple example

For a simple example, take the triangle bounded by $x=0,y=3x$ and $y=1$; scaling by $t$ just changes $y=1$ to $y=t$. The count is quasi-polynomial of period 3, but only has two distinct polynomials. The number of lattice points is

$$ 1+\frac{t^2+5t}{6}$$ if $t$ is congruent to 0 or 1 mod 3, and $$ 1+ \frac{t^2+5t-2}{6}$$ if $t$ is congruent to 2 mod 3.

Questions, general and specific

I'm interested in what we know about when or why these accidental equalities occur. That is a rather broad and open-ended question; so here's something a bit more specific.

If instead of just counting the lattice points in the $t$th dilate, we sum a polynomial function over the lattice points, we again get (quasi-)polynomial functions; call this Euler-Maclaurin theory. I had the naive hope that "Accidental" equalities in Ehrhart quasi-polynomials might extend to equalities in Euler-Maclaurin theory, but this appears not be always the case: if we try to sum the function $x$ over the lattice points in the $t$th dilate of the polytope in our example, we get $$\frac{t^3}{54}+\frac{t^2}{9}+\frac{t}{6}$$ for $t$ congruent to 0 mod 3, but $$\frac{t^3}{54}+\frac{t^2}{9}+\frac{t}{18}-\frac{5}{27}$$ for $t$ congruent to 1 mod 3.

If I have some rational polytope that has accidental equalities in its Ehrhart polynomials, and a specific polynomial I want to sum over it; is there some conditions in which the same accidental equalities will hold for the Euler-Maclaurin problem?

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This is an interesting and wide open question. The easiest case when considering Ehrhart quasipolynomials, namely, when all constituent polynomials are equal, goes by the name of period collapse. The most famous instance comes from representation theory (see this paper by J. De Loera & T. McAllister, which appeared in Discrete Comp. Geom. 32 (2004)). You might also find this paper by C. Haase & T. McAllister (which appeared in Cont. Math. 451 (2008)) illuminating.

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  • $\begingroup$ Thanks! The second paper looks like a great place to start from when I get the time to think about this again. $\endgroup$ – Paul Johnson Feb 6 '14 at 20:41

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