Thue equation

For other theorems named after Axel Thue, see Thue's theorem (disambiguation).

In mathematics, a Thue equation is a Diophantine equation of the form

$${\displaystyle f(x,y)=r,}$$

where ${\displaystyle f}$ is an irreducible bivariate form of degree at least 3 over the rational numbers, and ${\displaystyle r}$ is a nonzero rational number. It is named after Axel Thue, who in 1909 proved that a Thue equation can have only finitely many solutions in integers ${\displaystyle x}$ and ${\displaystyle y}$, a result known as Thue's theorem. ^[1]

The Thue equation is solvable effectively: there is an explicit bound on the solutions ${\displaystyle x}$, ${\displaystyle y}$ of the form ${\displaystyle (C_{1}r)^{C_{2}}}$ where constants ${\displaystyle C_{1}}$ and ${\displaystyle C_{2}}$ depend only on the form ${\displaystyle f}$. A stronger result holds: if ${\displaystyle K}$ is the field generated by the roots of ${\displaystyle f}$, then the equation has only finitely many solutions with ${\displaystyle x}$ and ${\displaystyle y}$ integers of ${\displaystyle K}$, and again these may be effectively determined. ^[2]

Finiteness of solutions and diophantine approximation

Thue's original proof that the equation named in his honour has finitely many solutions is through the proof of what is now known as Thue's theorem: it asserts that for any algebraic number ${\displaystyle \alpha }$ having degree ${\displaystyle d\geq 3}$ and for any ${\displaystyle \varepsilon >0}$ there exists only finitely many coprime integers ${\displaystyle p,q}$ with ${\displaystyle q>0}$ such that ${\displaystyle |\alpha -p/q|<q^{-(d+1+\varepsilon )/2}}$. Applying this theorem allows one to almost immediately deduce the finiteness of solutions. However, Thue's proof, as well as subsequent improvements by Siegel, Dyson, and Roth were all ineffective.

Solution algorithm

Finding all solutions to a Thue equation can be achieved by a practical algorithm, ^[3] which has been implemented in the following computer algebra systems:

• in PARI/GP as functions thueinit() and thue().
• in Magma as functions ThueObject() and ThueSolve().
• in Mathematica through Reduce[]
• in Maple through ThueSolve()

Bounding the number of solutions

While there are several effective methods to solve Thue equations (including using Baker's method and Skolem's p-adic method), these are not able to give the best theoretical bounds on the number of solutions. One may qualify an effective bound ${\displaystyle C(f,r)}$ of the Thue equation ${\displaystyle f(x,y)=r}$ by the parameters it depends on, and how "good" the dependence is.

The best result known today, essentially building on pioneering work of Bombieri and Schmidt, ^[4] gives a bound of the shape ${\displaystyle C(f,r)=C\cdot (\deg f)^{1+\omega (r)}}$, where ${\displaystyle C}$ is an absolute constant (that is, independent of both ${\displaystyle f}$ and ${\displaystyle r}$) and ${\displaystyle \omega (\cdot )}$ is the number of distinct prime factors of ${\displaystyle r}$. The most significant qualitative improvement to the theorem of Bombieri and Schmidt is due to Stewart, ^[5] who obtained a bound of the form ${\displaystyle C(f,r)=C\cdot (\deg f)^{1+\omega (g)}}$ where ${\displaystyle g}$ is a divisor of ${\displaystyle r}$ exceeding ${\displaystyle |r|^{3/4}}$ in absolute value. It is conjectured that one may take the bound ${\displaystyle C(f,r)=C(\deg f)}$; that is, depending only on the degree of ${\displaystyle f}$ but not its coefficients, and completely independent of the integer ${\displaystyle r}$ on the right hand side of the equation.

This is a weaker form of a conjecture of Stewart, and is a special case of the uniform boundedness conjecture for rational points. This conjecture has been proven for "small" integers ${\displaystyle r}$, where smallness is measured in terms of the discriminant of the form ${\displaystyle f}$, by various authors, including Evertse, Stewart, and Akhtari. Stewart and Xiao demonstrated a strong form of this conjecture, asserting that the number of solutions is absolutely bounded, holds on average (as ${\displaystyle r}$ ranges over the interval ${\displaystyle |r|\leq Z}$ with ${\displaystyle Z\rightarrow \infty }$). ^[6]

See also

• Roth's theorem
• Faltings's theorem

References

1. A. Thue (1909). "Über Annäherungswerte algebraischer Zahlen". Journal für die reine und angewandte Mathematik . 1909 (135): 284–305. doi:10.1515/crll.1909.135.284. S2CID   125903243.
2. Baker, Alan (1975). Transcendental Number Theory. Cambridge University Press. p. 38. ISBN   0-521-20461-5.
3. N. Tzanakis and B. M. M. de Weger (1989). "On the practical solution of the Thue equation". Journal of Number Theory . 31 (2): 99–132. doi: 10.1016/0022-314X(89)90014-0 .
4. E. Bombieri and W. M. Schmidt (1987). "On Thue's equation". Inventiones Mathematicae . 88 (2): 69–81. Bibcode:1987InMat..88...69B. doi:10.1007/BF01405092. S2CID   119634267.
5. C.L. Stewart (1991). "On the number of solutions to polynomial congruences and Thue equations". Journal of the American Mathematical Society . 4 (4): 793–835. doi: 10.2307/2939289 . JSTOR   2939289.
6. C.L. Stewart and Stanley Yao Xiao (2019). "On the representation of integers by binary forms". Mathematische Annalen . 375 (4): 133–163. arXiv: 1605.03427 . doi: 10.1007/s00208-019-01855-y .

Further reading

• Baker, Alan; Wüstholz, Gisbert (2007). Logarithmic Forms and Diophantine Geometry. New Mathematical Monographs. Vol. 9. Cambridge University Press. ISBN   978-0-521-88268-2.