218 - Österreichische Mathematische Gesellschaft

time-consuming is the fact that a full search for multivariate problems in integers
consumes exponentially more time with each variable and size of the domain to
be explored.
The first breakthrough at this bottleneck is provided by a generic algorithm – q-
class decomposition – which allows the computation of resonance solutions for
arbitrary irrational dispersion functions. The algorithm is based on number-theoretical
results most important of them being a linear independence of algebraic
numbers from different expansions of the field of rational numbers. Compared to
the brute-force approach, q-class decomposition gives enormous computational
advantages. For instance, all the solutions of the (25) in the computation domain
|m|,|n| ≤ 128 were found within 2.5 sec, while for |m|,|n| ≤ 1000 it takes about
12 minutes [10].
The main idea of the q-class decomposition is to divide an integer lattice {(m,n)}
into a set of sublattices, according to the necessary condition for existence of
the solutions for a Diophantine equation with irrationalities. This condition will,
of course, depend on the form of equation and the form of the irrationalities.
However, the general scheme is the same. To illustrate this, we define below q-
classes and formulate corresponding necessary conditions for the (25).
Definition. Let (m,n) be a node of an integer lattice, m,n ∈ Z, the index q of the
node (m,n) is an integer which is not divisible by the fourth power of any prime,
i.e.
q = p α 1
1 ··· pα s
s , α j ∈ [1,2,3] (27)
with distinct primes p j . The set of all nodes with the same index q is called the
q-class and denoted by Cl q .
Necessary conditions. If four nodes (m i ,n i ), i = 1,...,4 form a solution of the
(25), one of two conditions has to be satisfied:
• ∃q : (m i ,n i ) ∈ Cl q , ∀i = 1,...,4; or
• ∃q 1 ,q 2 : (m 1 ,n 1 ),(m 3 ,n 3 ) ∈ Cl q1 & (m 2 ,n 2 ),(m 4 ,n 4 ) ∈ Cl q2 .
This means that the search for solutions on the integer lattice {(m,n)} is now
reduced to the search for solutions in sublattices generated by only one or two
q-classes; this reduces drastically the computational time. Another substantial
improvement is due to the theorem of Gauss on the decomposition of an integer
into a sum of two squares – some of the q-classes turn out to be empty.
3 Nonlinear resonance analysis
The development of the general method for computing resonances opened up a
variety of new avenues for further theoretical investigations. In the course of the
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