Erdős 699: limits of adding polynomial factors

This is a retained earlier research note. The current reading guide gives the latest status and distinguishes completed proofs from remaining cases.

P699 research · Erdős problem 699

For three remaining cofactor profiles, adding more polynomial factors cannot make the usual asymptotic dimension count sufficient. This statement includes factors of arbitrary degree, not just lines or conics. It identifies where new arithmetic information is needed; it neither supplies bad triples nor rules out other interpolation arguments.

The profiles are listed in increasing prime order. A coordinate $e_p$ at position $r_p$ gives the row capacity $\alpha_{r_p}=e_p$; all other row capacities are 1.

These statements concern the displayed profiles. Stronger cofactor inequalities or an argument excluding the corresponding position assignment can remove a profile from consideration.

A weighted Bezout lemma

Let

Suppose nonnegative rational point weights $\lambda_P$ satisfy

and every affine line contains point weight at most 1. Suppose also that there are affine lines $L_j$ and nonnegative rational weights $w_j$ such that

Then every nonzero polynomial $F\in\mathbf Q[X,Y]$ satisfies

To prove this, first remove every affine-line component of $F$. Such a component already satisfies the assertion by the line-weight hypothesis. For the remaining polynomial $G$, no covering line $L_j$ is a component, so Bezout gives

Multiply by $w_j$, sum, and use the pointwise covering inequality. Add back the removed line components. The argument applies without a symmetry assumption on $F$.

Consequence for prescribed factors and residual jets

Allow any collection of factors $F_k$, with nonnegative normalized powers $c_k$, and prescribed row multiplicities $m_t$. Let $\rho_P\ge0$ be residual jet multiplicities satisfying

If the remaining degree is normalized to 1, the arithmetic gain needed for the usual interpolation argument is

The lemma implies $\sum_P\lambda_P\rho_P\ge1$. Therefore

For each profile above, the denominator is strictly below 1. Thus no choice of factors, factor degrees, or real multiplicities can give the sufficient asymptotic dimension inequality $\sum_P\rho_P^2<1$ for that profile. This does not exclude a construction using extra dependencies among jet conditions, another polynomial existence theorem, or additional arithmetic constraints.

Exact certificates

The rational certificate data record all point weights and line covers. Their squared norms and cover weights are respectively

The standalone checker uses rational arithmetic only. It checks the row sums, every line through two grid points, every pointwise covering inequality, and both strict bounds. Lines containing at most one grid point automatically have weight at most 1, since each row sum is at most 1. Run it with

.venv/bin/python wiki/research/erdos_699/computations/erdos_699_polynomial_factor_barriers.py

The exploratory search leading to this lemma considered symmetric cubics through five off-diagonal point orbits, cubics singular at a diagonal point, and divided differences $(f(X)-f(Y))/(X-Y)$ for polynomials $f$ with distinct integer roots in $[0,i-1]$. Those searches are unnecessary for verifying the result: the rational line covers handle every polynomial degree at once. The analogous cover feasibility calculation did not produce such a certificate for the tested residual index-12 profile; no conclusion about that profile follows from this note.