Only h_0=H(0) is needed. Let V_0 replace V's constant-coefficient column by b=Omega\*y. Cramer's rule gives h_0=det(V_0)/(Omega\*det(V)), with det(V) nonzero. Use fraction-free Gaussian elimination, not permutation enumeration or floating-point arithmetic, to compute the determinants. The standard method uses O(K^3) integer arithmetic operations, with row pivoting when necessary and exact divisions. Its retained entries are signed minors; intermediate products and differences still have polynomial bit length by the minor bound. The numerator matrix may be singular, yielding a zero answer. For clause-indicator factors, H(0) is an integer count, so the final division is exact.

Recovering the constant coefficient with bounded exact arithmetic

A k-by-k determinant is a signed sum of k! products, each choosing one entry from every row and column. A minor is the determinant of a square submatrix. If an entire K-by-K integer matrix has entries of magnitude at most 2^B, every minor has magnitude at most K!\*2^(K\*B), hence O(K\*B+K\*log2(K+1)) bits. This bounds the objects used by fraction-free elimination. It does not suggest computing determinants by enumerating all permutations; that would be factorial work. For the Hermite matrix augmented by the integer data column, use a B large enough for both the matrix entries and that data.

AI Agents

Let K=Q\*J and pad the unknown polynomial coefficient list to degrees 0,...,K-1. The local coefficient [z^j]T^u at T=t+z is binom(u,j)\*t^(u-j) for u>=j, otherwise zero. These integers form a K-by-K matrix V, one row per (t,j). The previously proved Hermite uniqueness makes V nonsingular. Its entries have O(K\*log2(Q+2)) bits: binom(u,j)<=2^u and t<=Q. If the computed local data y share denominator Omega, then b=Omega\*y is integral. Solving V\*(Omega\*h)=b recovers the polynomial coefficients h without floating-point interpolation.

The exact integer matrix for local-coefficient interpolation

The determinant of a matrix, $A$, is the product of $A$'s eigenvalues, where $A$ is an $m x m$ matrix. It has numerous applications in finding volume and using integrals. It can only be used on square matrices.

Determinants of different sizes:
$det(\begin{bmatrix} a \end{bmatrix}) = a$
$det(\begin{bmatrix} a & b\\ c & d \end{bmatrix}) = ad-bc$
$det(\begin{bmatrix} a & b & c\\ d & e & f\\ g & h & i \end{bmatrix}) = a(ei-fh) - b(di-fg) + c(dh-eg)$

Linear Algebra (Determinant)

OpenAI. The Exponential Time Hypothesis is False. Manuscript dated October 17, 2026, revised October 25, 2026. This reference pins commit 42907c9e999cc00d6abe7a50bfc1023250edd1b9. The introduction defines the exact-counting task used by this learning path; citing it does not establish that its main theorem is correct.

https://github.com/openai/math/blob/42907c9e999cc00d6abe7a50bfc1023250edd1b9/preprints/The-Exponential-Time-Hypothesis-is-False-October-25-2026/paper.pdf

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