Erdős 699: why more curve factors do not settle one index-six case

P699 research · The full-overlap column classes · The index-eight degree bound

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Consider the index-six profile

$$(r_2,r_3,r_5)=(2,1,0),\qquad (e_2,e_3,e_5)=(.31,.36,.31).$$

Its row capacities are $\alpha=(.31,.36,.31,1,1,1)$.

Proposition. If a nonzero polynomial $F\in\mathbf C[X,Y]$ has multiplicity at least $m_r$ at every nonnegative integer point on row $X+Y=r$, $0\le r\le5$, then

$$\boxed{\deg F\ge \frac{31}{100}m_0+\frac9{25}m_1+\frac{31}{100}m_2+m_3+m_4+m_5.}\tag{1}$$

This holds in every degree, without requiring symmetry, a coprime polynomial family, or full rank of a jet matrix. In particular, allowing common curve zeros whose bad points can be bounded by Runge's theorem, positivity, or a local argument does not supply the strict weighted-degree gain at this profile: inequality (1) holds for each polynomial itself.

The proposition does not say that any bad triple realizes this profile. Stronger cofactor estimates, additional restrictions on the Kummer labels, and sharper bounds for polynomial values at a particular column remain outside its scope.

Point weights and a nine-line covering

Put $\Delta_6=\{(x,y)\in\mathbf Z_{\ge0}^2:x+y\le5\}$. Give these points weights $w_{x,y}$ using the following numerators, all divided by $16000$. Within row $r$, the entries are ordered by $x=0,\ldots,r$.

Their row sums are exactly $\alpha_r$. Give the following lines coefficients $\lambda_L$ with the same denominator $16000$.

The exact incidence inequalities are

$$w_P\le\sum_{L\ni P}\lambda_L,\qquad \sum_L\lambda_L=\frac{7927}{8000}<1.\tag{2}$$

Every covering line itself has weighted incidence at most one: $X=2$ and $Y=2$ have incidence $7179/8000$, and the other seven have incidence one.

For an irreducible curve $C$ different from these nine lines, Bézout gives

$$\sum_{P\in C\cap L\cap\Delta_6}\operatorname{mult}_P C\le\deg C$$

for every covering line. Multiplying by $\lambda_L$, summing, and using (2) proves

$$\sum_{P\in\Delta_6}w_P\operatorname{mult}_P C \le\frac{7927}{8000}\deg C<\deg C.$$

The covering lines satisfy the non-strict bound separately. Since multiplicities and degrees add under factorization, every nonzero polynomial consequently satisfies

$$\sum_{P\in\Delta_6}w_P\operatorname{ord}_P F\le\deg F.\tag{3}$$

Taking the weighted sum of its row multiplicities proves (1).

The standard-library checker and certificate verify all 21 point weights, all nine line incidences, and the row sums using exact arithmetic. No optimization solver or bounded-degree search is needed to verify the proof.