Erdős 699: an all-degree polynomial obstruction at index eight

P699 research · The remaining cofactor profile · The corresponding quintic obstruction

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Consider the remaining index-eight profile

$$(r_2,r_3,r_5,r_7)=(1,3,0,2),\qquad (e_2,e_3,e_5,e_7)=(.3,.3,.3,.3).$$

Its row capacities are

$$\alpha=(.3,.3,.3,.3,1,1,1,1).$$

Proposition. If a nonzero polynomial $F\in\mathbf C[X,Y]$ has multiplicity at least $m_r$ at every nonnegative integer point on row $X+Y=r$, $0\le r\le7$, then

$$\boxed{\deg F\ge\frac3{10}(m_0+m_1+m_2+m_3)+m_4+m_5+m_6+m_7.} \tag{1}$$

This holds in every degree and does not assume symmetry, coprime sections, or surjectivity of a jet matrix. Thus the strict weighted-degree gain required by the present polynomial divisor argument cannot hold at this profile. Finding a polynomial whose common curve zeros are effectively bounded, through Runge's theorem or a local nonvanishing argument, does not by itself change (1).

The statement does not assert that the profile is realized by any bad triple. Stronger cofactor estimates, restrictions on the actual Kummer labels, or sharper estimates for polynomial values at the particular column remain outside this obstruction.

Point weights

Let $\Delta_8=\{(x,y)\in\mathbf Z_{\ge0}^2:x+y\le7\}$. Give its points weights $w_{x,y}$ as follows. The entries are numerators with denominator 100; within row $r$, they are ordered by $x=0,\ldots,r$.

Their row sums are exactly $\alpha_r$. We prove the stronger universal inequality

$$\sum_{P\in\Delta_8}w_P\operatorname{ord}_P F\le\deg F. \tag{2}$$

A thirteen-line covering

Assign the following coefficients $\lambda_L$ to lines. Again, all numerators have denominator 100.

The exact incidence checks give

$$w_P\le\sum_{L\ni P}\lambda_L,\qquad \sum_L\lambda_L=\frac{49}{50}<1. \tag{3}$$

Each covering line itself has weighted incidence at most one: $X=2,Y=2$ have incidence $99/100$; $X=3,Y=3$ have incidence $91/100$; the other nine lines have incidence one.

Let $C$ be an irreducible curve different from these thirteen lines. Bézout's theorem gives

$$\sum_{P\in C\cap L\cap\Delta_8}\operatorname{mult}_P C\le\deg C$$

for every covering line. Combining these inequalities with (3) yields

$$\sum_{P\in\Delta_8}w_P\operatorname{mult}_P C \le\frac{49}{50}\deg C<\deg C.$$

The covering lines obey the non-strict bound separately. Factoring an arbitrary nonzero polynomial into irreducible factors therefore proves (2). Its row-multiplicity consequence is (1).

The integer/rational checker and certificate verify all 36 point weights, all thirteen line incidences, and the row sums. No optimization solver or bounded-degree search is needed to verify the proof.