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In the graded situation the concept of "minimal generators" is well-defined. Just think about the minimal generators as part of the minimal free resolution. Their number is the first total Betti num …

What is going on can be explained completely in terms of exponent vectors. To do so we can map each generator $x^u -x^v$ of $J$ to the exponent vector $u-v \in \mathbb{Z}^n$. Conversely, for each inte …

One general fact that comes to mind: If an ideal $I\subset \mathbb{k}[x_1,\dots,x_n]$ contains an element of the form $f = gx_1 + h$ where $g,h$ don't use $x_1$, and $g$ is a nonzerodivisor mod $I$, …

Let $I$ be the ideal generated by your binomial equations after clearing denominators. As a general rule with binomial equations, disregard coefficients in the first run (see here for why and how: ht …

Many introductory texts on algebraic geometry set up some sort of algebra-geometry dictionary in which radical ideals correspond to varieties, and so on. I am wondering if there is a geometric way to …