Put $R_n=\mathbb{R}[t_0,\dotsc,t_n]/(\sum_it_i-1)$ (the ring of polynomial functions on the $n$-simplex). Consider a monomial $t^a=t_0^{a_0}\dotsb t_n^{a_n}$. Let $(b_0,\dotsc,b_n)$ be the sequence $(a_0,\dotsc,a_n)$ arranged in nondecreasing order. I'll say that $t^a$ is admissible if $b_n=b_{n-1}+1$. One can show that the admissible monomials form a basis for $R_n$ with various convenient properties. In the case $n=1$, you just get the monomials $t_0^{i+1}t_1^i$ and $t_0^it_1^{i+1}$, for example (so $1$ has to be expressed as $t_0+t_1$). Does this appear in the literature, and if so, under what name?
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asked May 3, 2013 at 13:30
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$\begingroup$ Rings of this type, generalized to arbitrary integral polytopes and permitting other coefficients than 1, are considered (at length) in Positive polynomials and product type actions of compact groups (AMS Memoirs, 320, 1985) and especially Positive Polynomials, Convex Integral Polytopes, and a Random Walk Problem (SLN 1282, 1987), both by me (there are also later publications dealing with them). But they don't address your specific problem. $\endgroup$– David HandelmanCommented Jun 22, 2016 at 11:55
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