This paper proves a sharp negative result about Hilbert’s tenth problem over the integers. The author shows there is no algorithm that, given any polynomial with integer coefficients in ten unknowns, can always decide whether that polynomial has an integer solution. This improves an earlier result that needed eleven unknowns and therefore lowers the number of variables known to be enough to force undecidability.

Hilbert’s tenth problem asks for a single mechanical procedure to test whether a given polynomial equation with integer coefficients has an integer solution. In 1970 Yuri Matiyasevich, completing work of Davis, Putnam and Robinson, showed there is no such procedure in general: certain sets of integers that can be generated by algorithms (called recursively enumerable or r.e. sets) can be encoded by polynomial equations. Some r.e. sets are not decidable, so no uniform algorithm can work for all polynomials. The new paper builds on that foundation to make the encoding work with only ten unknowns.

What the author does is construct, for any r.e. set A, an explicit integer polynomial P_A(a,z1,…,z10) with the property that the integer a lies in A exactly when there exist integers z1,…,z10 making P_A(a,z1,…,z10)=0. The paper says these polynomials are “effectively constructible,” meaning there is a rule to produce the polynomial from A. To reach this result the proof combines several number-theory tools and constructions listed in the paper: Lucas sequences (number sequences defined by a simple two-term recurrence, like a generalized Fibonacci sequence), valuations (which count how many times a prime divides a number), and a series of technical building blocks the author calls source-coding, interpolation, a divisibility–nonvanishing merger, a modified Lucas certificate, and a binomial-coefficient bridge. These parts are used to encode computable information into the arithmetic of integers via a single polynomial in ten variables.