Erdős 699: a smaller coefficient bound at index 20

P699 research · Effective bounds at 15 and 20 · The interpolation nonvanishing lemma

Complete exclusions. Indices $15,20$ are fully excluded by the cutoff-15 proof; the bounds and constructions below are retained as earlier stages and inputs.

The existing index-20 interpolation construction gives the much smaller algebraic tail

once its cofactor inequalities hold. This replaces the generic coefficient estimate that gave $10^{4000000}$. It is not an assertion that every source threshold is below $10^{6396}$: the original near-limit asymmetric estimates still have larger numerical thresholds. The polynomial bound below can also be combined with another certified cofactor gain for the same row weights.

An explicit complete kernel basis

Keep the index-20 vectors

and residual multiplicities $R_{xy}=\max\{0,m_{x+y}-h_x-h_y\}$ at $x,y\ge0$, $x+y<20$. The previously certified symmetric degree-96 jet map is surjective, of rank 2399. There are 2450 symmetric monomials through degree 97, so its degree-97 kernel has dimension 51.

The compressed coefficient artifact contains 51 integer polynomials in the basis $X^aY^b+X^bY^a$ for $a<b$ and $X^aY^a$ for $a=b$. The direct verifier checks:

• All 2399 required jets of every polynomial are exactly zero over the integers.
• The coefficient minor on the last 51 symmetric monomials has determinant $13345\pmod{65521}$, so the polynomials are independent.
• Their largest coefficient has 121 decimal digits, and their largest coefficient norm has 123 decimal digits.

Thus these polynomials form a rational basis of the complete symmetric degree-97 kernel. The existing multiplication-by-$X+Y$ lemma guarantees that they have no common zero when $X+Y>19$.

The basis was obtained by an exact Dixon solve for the 51 free columns, followed by primitive normalization, saturation of the integer lattice, and LLL reduction. The initial primitive basis had a 1696-digit maximum coefficient norm. Saturation uses the row Hermite form of the transpose: if $B$ has the basis vectors as rows and $H$ is a row basis of the coordinate lattice of $B^{\mathsf T}$, then $B^{\mathsf T}H^{-1}$ is integral and generates the saturated column lattice. Exact identities verify this step and the unimodular LLL transformation. Verification of the retained result does not depend on any property of the search or reduction algorithm: integer jets and the nonzero coefficient minor suffice.

Evaluating the polynomials on a large row

For each retained polynomial $G$, write its degree-97 part as

The exact largest Bernstein coefficient, across the entire basis, is

For $X,Y\ge0$ and $X+Y=n$, the binomial theorem gives $|G_{97}(X,Y)|\le Bn^{97}$. All lower-degree terms together have magnitude less than $n^{97}$ when $n\ge10^{123}$, because the full coefficient norm of every $G$ is less than $10^{123}$. Consequently

This bound uses only the nonnegative physical coordinates, not a bound for arbitrary complex evaluations.

The weighted divisibility constant

Set

Here $2\sum h_a=466$. Pairing the factors gives the sharper direct bound

The exact off-maximum small-prime constant is

For the normalized cofactor exponent $v_p$, retain

at singleton positions instead of replacing the last factor by $2^{v_p}$. At a collision position the selected prime powers have product at most $n$. Write $L=\sum t m_t=7692$. The numerator factors and all singleton normalization factors together are at least

This follows by expanding products of factors in $[0,1]$ and bounding the total singleton exponent by $8m_0$.

For every bad pair beyond the cofactor thresholds, select a retained $G$ with $G(j,c)\ne0$, which is possible by nonvanishing. Apply weighted Kummer divisibility to $HG$. The already certified uniform gain is $713/5000$, so

The second inequality is checked by an integer comparison. Finally, $6396\cdot713/5000>912$. Since $10^{6396}$ also exceeds the elementary and evaluation thresholds, the claimed algebraic tail follows.

Retained verification

The small JSON report contains the coefficient norm, Bernstein bound, coefficient-artifact checksum, independence minor, exact off-maximum factors, and the resulting exponent comparison. Run the checker with the pinned dependencies installed in the repository's .venv/:

.venv/bin/python wiki/research/erdos_699/computations/erdos_699_index_20_small_height.py

Ordinary verification checks the retained integer polynomials directly. The optional --regenerate flag repeats the exact solve, saturation, and LLL construction. No binomial rows are enumerated.