I am interested in polynomials with few terms ("short polynomials", "fewnomials") in ideals. A simple to state question is


Given an ideal $I\subset k[x\_1,\dots,x\_n]$, what is the shortest polynomial in $I$?


There are answers for "Does an ideal contain a monomial/binomial?", but the general question seems to be hard. Let's try an easy specific question:


How few terms can a polynomial in $\langle(x-1)(y-1)(x^2y-1)\rangle \subset k[x,y]$ have?


The generator has 8 terms so this is an upper bound for the minimum. A lower bound for the minimum is 6 which can be seen as follows: Let $f$ be the generator. The Newton polytope of $f$ is a hexagon with two interior points. Any polynomial in $I$ is of the form $fg$ for some $g\in k[x,y]$ and its Newton polytope is $\text{Newton}(f) + \text{Newton}(g)$. Since the number of vertices cannot decrease under the Minkowski sum, $fg$ has at least six terms.


Are there polynomials with six or seven terms in $I$?



What is the shortest polynomial divisible by $(x-1)(y-1)(x^2y-1)$
A simpler answer is given by $x^4y^2 - x^4y - x^2y^2 + x^2 + y - 1=(x+1)f$, where $f$ is the generator of the ideal.


