1 · The essentials
Can there be a method that decides, for every polynomial equation with integer coefficients, whether it has a solution in rational numbers—that is, fractions of integers? The main manuscript claims a negative answer. It states that no algorithm can give the correct answer for all such equations. The number of unknowns can vary and is part of the input. This claim concerns a universal method, not the impossibility of solving any particular equation. A companion manuscript presents a result about a family of curves, but the abstracts do not specify its role in the main argument. The main result is not formalized in Lean and has not been formally verified.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine a machine that receives an equation and must always answer correctly either “a solution in fractions exists” or “none exists.” The manuscript claims that such a universal machine cannot exist, even though individual equations may be handled separately.
Why it matters
The claimed conclusion would set a precise limit on what a general procedure can decide about equations with rational solutions. It would give a negative answer to Hilbert’s tenth problem over the rationals as formulated in the source. This manuscript claim must, however, be distinguished from formal verification, which has not been carried out.
2 · Going further
The problem takes a polynomial as input, with itself part of the input. It asks whether there is a tuple such that . The main manuscript claims that no algorithm decides this question for every input. Its abstract states the conclusion but describes neither a construction nor the steps of the argument.
The companion manuscript concerns elliptic curves over with a nonzero rational point of order two. It claims a pointwise “2-converse”: if the -Selmer corank is zero or one, then the analytic rank and the Mordell–Weil rank equal that corank, and the Shafarevich–Tate group is finite. The abstract explicitly allows arbitrary reduction at 2.
The two texts are presented as belonging to the same family, but their abstracts give no explicit logical connection between the results. They also do not supply the main ideas of the arguments, so these materials alone do not explain how the elliptic-curve result might contribute to the conclusion about polynomial equations. The main result is neither formalized in Lean nor formally verified.
Useful notions
- Integers and rational numbers
- Polynomials in several unknowns
- Algorithms and decision problems
- Basic notions of elliptic curves
What was known before / what is new
Before
The source identifies the question as Hilbert’s tenth problem over the rational numbers. It supplies neither historical background nor earlier results that would place the two claims in a more detailed context.
New
The main manuscript claims a negative answer to the decision problem when the number of variables is part of the input. The companion manuscript claims a 2-converse for elliptic curves with a nonzero rational point of order two, assuming that the Selmer corank is zero or one. This second statement allows arbitrary reduction at 2.
Limits and assumptions
The main statement concerns integer-coefficient polynomials, rational solutions, and a number of variables that is part of the input; the abstract gives no separate conclusion for a fixed number of variables. The companion result assumes nonzero rational two-torsion and a -Selmer corank of zero or one; it places no restriction on reduction at 2. The abstracts describe neither the arguments nor their connection and do not specify which questions remain open. The main result is not formalized in Lean and has not been formally verified.
Mini glossary
- Rational number —
- A number that can be written as a fraction of two integers with a nonzero denominator.
- Rational zero —
- A choice of rational values for all variables that makes the polynomial equal to zero.
- Decision problem —
- A question requiring a yes-or-no answer for each input.
- Nonzero rational point of order two —
- A point on an elliptic curve defined over the rationals, distinct from the identity point, whose sum with itself is the identity point.
Keywords : Hilbert’s tenth problem · Rational numbers · Elliptic curves
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Proves that no algorithm decides whether an integer-coefficient polynomial in an arbitrary number of variables has a rational zero, resolving Hilbert's tenth problem over ℚ negatively.
To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.
Manuscripts in this family
Hilbert’s tenth problem over the rational numbers
06/10/2026We give a negative answer to Hilbert's tenth problem over the rational numbers: no algorithm decides whether a polynomial with integer coefficients has a rational zero. The number of variables is part of the input.
A pointwise 2-converse for elliptic curves with rational two-torsion
07/10/2026We prove a pointwise 2-converse for elliptic curves over with nonzero rational two-torsion: if the -Selmer corank is zero or one, then the analytic rank and Mordell–Weil rank equal that corank, and the Shafarevich–Tate group is finite. The result allows arbitrary reduction at 2.