Gupta's bounded transcendence-degree conjecture for polynomial sets

Let $\mathcal{F}_1,\ldots,\mathcal{F}_k$ be finite sets of irreducible homogeneous polynomials in $\mathbb{C}[x_1,\ldots,x_n]$ of degree at most $r$, with $\bigcap_i\mathcal{F}_i=\emptyset$. Suppose that for every $k-1$ polynomials $Q_1,\ldots,Q_{k-1}$, each chosen from a distinct set, there are $P_1,\ldots,P_c$ in the remaining set such that whenever $Q_1,\ldots,Q_{k-1}$ vanish, the product $\prod_{i=1}^cP_i$ also vanishes.

Gupta's bounded transcendence-degree conjecture. There is a function $\lambda$ such that

$$\operatorname{trdeg}_{\mathbb{C}}\left(\bigcup_i\mathcal{F}_i\right)\leq\lambda(k,r,c).$$

Here transcendence degree is the same as algebraic rank; equivalently, the condition says $\prod_{i=1}^cP_i\in\sqrt{(Q_1,\ldots,Q_{k-1})}$. The conjecture generalizes the relevant Sylvester–Gallai phenomena from linear to bounded-degree polynomials and was open in the source for degrees greater than one, while the paper establishes related quadratic cases.