A while ago I asked a question at Mathematica.SE about how to factorize a polynomial into terms with as few monomials as possible each. I now realized that I actually do not know what is rigorous mathematics behind this.
In fact that one was about univariate polynomials, and the same can be asked about several variables.
The question is whether there is some rigorous mathematics (algebra/geometry) behind asking for simultaneously minimizing the number of factors, and the number of monomials in each factor.
For an illustration, here is an example from that question: factorization of $$1 - q^8 - q^{11} - q^{14} + q^{19} + q^{22} + q^{25} - q^{33}$$into $\mathbb Q$-irreducibles is \begin{multline*}(1 - q)^3 (1 + q)^2 (1 + q^2) (1 + q^4) (1 - q + q^2 - q^3 + q^4 - q^5 + q^6)\\ (1 + q + q^2 + q^3 + q^4 + q^5 + q^6)\\ (1 + q + q^2 + q^3 + q^4 + q^5 + q^6 + q^7 + q^8 + q^9 + q^{10})\end{multline*} while "my" optimal expression would be$$(1 - q^8) (1 - q^{11}) (1 - q^{14}).$$
For many variables, and also allowing rational expressions and residues, one might ask for expressionsI would like$$F=\frac{P_1\cdots P_k}{Q_1\cdots Q_l}+R_1\cdots R_m$$with each of $P$, $Q$, $R$ having as few monomials as possible. For example, given$$F(x,y)=x^{14}+2 x^{12} y^3+3 x^{10} y^6+4 x^8 y^9+4 x^6 y^{12}+3 x^4 y^{15}+x^2 y^{18}+y^{21}$$how to arrive at$$F(x,y)=\frac{\left(x^8-y^{12}\right) \left(x^{10}-y^{15}\right)}{\left(x^2-y^3\right)^2}-\left(x y^9\right)^2$$recognize in\begin{multline*}\left(x+y^3\right)^2 \left(x^2+y^6\right)^2 \left(x^2-x y^3+y^6\right) \left(x^2+x y^3+y^6\right) \left(x^4+y^{12}\right)\\ \left(x^4-x^2 y^6+y^{12}\right) \left(x^8+y^{24}\right)\end{multline*}that it is equal to$$\frac{\left(x^{12}-y^{36}\right) \left(x^{16}-y^{48}\right)}{\left(x-y^3\right)^2}$$