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Assume for simplicity that your polynomials are homogeneous of degree three. If you have a complete intersection, it has Hilbert series $$ (1-t^3)^{18}/(1-t)^{19} = (1+t+t^2)^{18}/(1-t) $$ It is conc …

There seems to be some terminology drift here. I would say that "order" would be called degree in modern terminology, for example. Here is the way I see it, and please someone correct me if I am wron …

I don't have a complete solution, but the following may be helpful. Change variables by $z_i = \sum_j y_j \xi^{ij}$ where $\xi$ is $m$-th primitive root of $1$. Then the first line equations (I am us …

Clearly, the statement is invariant under multiplication by $a\in K$, so we may assume that $W\ni 1$. This implies that $W\supseteq R$, and we want to show that $W=S$. Suppose that $t\in W$. I claim …

Let's see what happens in dim 2. You have $conv((0,0),(ra_1+sb_1,0),(0,ra_2+sb_2))$. The number of points in the closed triandle $(0,0),(A,0),(0,B)$ is $(A+1)(B+1)/2$ plus half the number of points on …

In my paper arXiv:math/9802052 I give an argument in the graded case. The general case can be approached similarly (as sketched in the paper). The sequence in question is made from log-derivatives of …