Robust Radical Sylvester-Gallai Theorem for Quadratics

NEW

Problem: Reconstructed literal conjecture: for every $k,d,c\in\mathbb N^*$ there is a bound $B(k,d,c)$ such that, over a polynomial ring over a field, any finite sets $F_1,\dots,F_k$ of irreducible polynomials of degree $\le d$ satisfying

and such that for every choice of $k-1$ polynomials from distinct $F_i$’s, the remaining set contains $P_1,\dots,P_c$ with

has $\operatorname{trdeg}(\bigcup_i F_i)\le B(k,d,c)$.

This is the statement supported by the quoted Conjecture 1.5. The base field and scalar-associate conventions are omitted; the counterexample below works over $\mathbb C$ and does not rely on scalar multiples.

Result: The conjecture as stated is false already for $(k,d,c)=(4,1,1)$.

For $n\ge1$, work in

Let

Each element is irreducible of degree $1$. Also

since no $x_i$ equals any $y_j$.

Check the radical condition. If the remaining set is $F_1$, then the chosen three polynomials come from $F_2,F_3,F_4$, say $x_a,y_b,y_c$. Since $x_a\in F_1$ and

the condition holds. The same argument works for remaining $F_2$. If the remaining set is $F_3$, one of the chosen polynomials comes from $F_4$, say $y_c$, and $y_c\in F_3\cap \sqrt{(\cdots,y_c)}$. Similarly for $F_4$.

Thus the hypotheses hold with $k=4,d=1,c=1$. But

which is unbounded as $n\to\infty$. Therefore no function $B(4,1,1)$ exists.

The defect is the missing non-overlap condition between color classes. A natural repair would require the $F_i$’s to be pairwise disjoint up to scalar associates, or otherwise impose a minimality condition excluding duplicated color classes.

Citation: No external citation is needed; the counterexample is elementary and given above.