This is, probably, a question for those knowledgeable on the subject of Brion's theorem and its applications.

Lately, I've been dealing with situations of the following sort. Suppose we are given a polytope $P\subset\mathbb{R}^n$ as well as a linear map $\varphi:\mathbb{R}^n\rightarrow\mathbb{R}^m$ with $m<n$. Bases are fixed in both spaces and $\varphi$ maps integral points to integral points (in other words, has an integral matrix with respect to these bases). Also $P$ is rational and thus subject to Brion's theorem.

Now, $\varphi$ can be viewed as a change of variables $F$: $$F:x_i\rightarrow\prod\limits_{j=1}^m y_j^{\varphi_{ji}}, 1\le i\le n.$$ Here $x_i$ are the exponents of basis vectors in $\mathbb{R}^n$ (the variables in Brion's formula), $y_i$ -- the exponents of basis vectors in $\mathbb{R}^n$. Thus, here $\exp(\bar v)$ is simply substituted by $\exp(\varphi(\bar v))$.

In my case $F$ is applicable to the identity provided by Brion's theorem (the denominators do not vanish identically) and I noticed the following to hold. The summands (vertex cones' integer point transforms) corresponding to vertices of $P$ not mapped to vertices of $\varphi(P)$ vanish under $F$!

I see that the naive generalization is far from being correct... But isn't there, just maybe, a known theorem stating something of the sort?

The above is what I'm chiefly after. However, another interesting trait of my polytopes is that the summands which don't vanish, turn into neat products superficially resembling integer point transfroms of simplicial cones, namely $F$ transforms them into $$\exp(\varphi(v))\prod\limits_{i=1}^n \frac 1{1-\exp(\varphi(e_i))},$$ where $\{e_i\}$ is a subset of edges of the (non-simplicial) vertex cone at vertex $v$.

Any known reasons for such behaviour to take place? Unfortunately, I can't grasp any geometrical sense of the set $\{e_i\}$.

Update. Yes, here is a small nontrivial example. Pyramid with vertices $(0,0,0)$,$(2,0,0)$,$(0,2,0)$,$(2,2,0)$ and $(1,1,1)$, map $(a,b,c)\rightarrow(a,b)$ (projection onto the base). Then the integer point transform of the cone at the apex is $$f=\frac{(1+xy)(1-z^{-2})}{(1-xyz^{-1})(1-x^{-1}y^{-1}z^{-1})(1-xy^{-1}z^{-1})(1-x^{-1}yz^{-1})}$$ and $F$ is given by $x\rightarrow t,y\rightarrow w,z\rightarrow 1$. Thus, $f$ is seen to vanish under F.

Update 2. Actually, just like in the above example, each of the vertex cones $C$ the integer point transfrom of which vanishes, has two edges $e_1$ and $e_2$ such that $\varphi(e_1)=-\varphi(e_2)$. This forces $\varphi(C)$ to have an apex of positive dimension and sometimes gives me the feeling that the rest should be obvious via some formal argument...

Update 3. Maybe, I didn't make clear enough what $F$ is... The definition I give is a bit formal as compared to natural. The natural way might be to first introduce $F$, substituting each variable with a Laurent monomial in some other (smaller) set of variables. Then the matrix $φ$ can be defined via $F$.

And, just in case, in general, $\mathrm{IPT}(φ(C))\not=F(\mathrm{IPT}(C))$! (Integer Point Transform.)