hypothesisHypothesis claimClaim Index Reduction for Degenerated daesW. Yang,
W. Wu, and G. Reid ex_supplement

# Index Reduction for Degenerated Differential-Algebraic Equations by
Embedding and Real Numerical Algebraic Geometry ††thanks: Submitted to the
editors DATE. This work is partially supported by the projects of Chongqing
(2020000036, 2021000263,cstc2020yszx-jcyjX0005) and the National Natural
Science Foundation of China (11771421)

Wenqiang Yang Chongqing Key Laboratory of Automated Reasoning and Cognition,
Chongqing Institute of Green and Intelligent Technology, Chinese Academy of
Sciences. 🖂Corresponding Author<EMAIL_ADDRESS>Chongqing School,
University of Chinese Academy of Sciences. Wenyuan Wu22footnotemark: 2 Greg
Reid44footnotemark: 4 Mathematics Department, University of Western Ontario.

###### Abstract

To find consistent initial data points (witness points) for a system of
differential-algebraic equations, requires the identification of its missing
(hidden) constraints arising from differentiation of the system. An efficient
class of so-called structural methods exploiting a dependency graph for this
task was initiated by Pantiledes. The choice of graph is related to the choice
of ordering that minimize the solution dimension of the problem. However, this
method may fail. More complete methods rely on differential-algebraic geometry
but suffer from other issues (e.g. high complexity and instability on
approximate data). In this paper we give a new class of efficient structural
methods combined with new tools from numerical real algebraic geometry that
has much improved completeness properties. Existing structural methods may
fail for a system of differential-algebraic equations if its Jacobian matrix
after differentiation is still singular due to symbolic cancellation or
numerical degeneration. Existing structural methods can only handle
degenerated cases caused by symbolic cancellation. However, if a system has
parameters, then its parametric Jacobian matrix may be still singular after
application of the structural method for certain values of the parameters.
This case is called numerical degeneration.

In this paper, for polynomially nonlinear systems of differential-algebraic
equations, numerical methods are given to solve both degenerated cases using
numerical real algebraic geometry. First, we introduce a witness point method,
which produces at least one witness point on every constraint component
(manifold). This can help to ensure constant rank and detection of
degeneration on all components of such systems. Secondly, we present a
Constant Rank Embedding Lemma, then based on this lemma we propose an Index
Reduction by Embedding (IRE) method which can construct an equivalent system
with a full rank Jacobian matrix. Thirdly, IRE leads to a global structural
differentiation method, to solve degenerated differential-algebraic equations
on all components numerically. Application examples of models from circuits,
mechanics, and motion are used to demonstrate our method and its advantages.

###### keywords:

real algebraic geometry, constant rank, witness points, differential-algebraic
equations, structural methods.

68Q25, 68R10, 68U05

##  1 Introduction

Systems of differential-algebraic equations are widely used to model and
simulate dynamical systems such as mechanical systems, electrical circuits,
and chemical reaction plants [Ilchmann15]. We will often use the abbreviation
dae111A system of differential-algebraic equations will be denoted by dae
while daes will denote several such systems. for a system of differential-
algebraic equations. The name arose since such systems usually contain
differential equations with derivatives and algebraic equations without
derivatives. It was initially believed that any such dae could be easily
converted by coordinate changes and eliminations to a traditional explicit ode
\- the so-called underlying ode222A system of explicit ordinary differential
equations, in solved form for their highest derivatives, will be denoted by
ode.. However, they are now recognized as common and natural in applications,
to the extent that several modeling environments have them as their central
object. Indeed, the explicit underlying ode may be too expensive to un-cover,
and modern codes for numerical solution of a dae have made this unnecessary in
most cases. Even the idea there is a unique underlying ode is misleading since
the dae may yield several component manifolds with different behavior and
different underlying ode on each component.

daes are a subset of the set of general systems of partial differential
equations. The great geometer Cartan conjectured but was not able to prove
that after a finite number of prolongations (differentiations) of such
systems, they would become involutive, and a local existence and uniqueness
theorem could be stated for their solutions. Another greater geometer
Kuranishi eventually proved this result, albeit under conditions that are
difficult to render explicitly [Kuranishi57]. The number of prolongations
(differentiations) to uncover the underlying ode (i.e. the differential index)
corresponds to this result for daes; and it is equivalent to obtaining all the
constraints on initial data for existence and uniqueness of solutions. We note
that the differential index and underlying ode may be different on different
components of the dae.

One may try to numerically solve a dae directly without reducing its index by
prolongation. However, properly posing initial values for a dae requires that
they satisfy the missing constraints, and hence implicitly requires knowledge
of the prolonged form of the dae. This direct approach is prone to order
reduction, instability, inaccuracy and the tendency for the approximate
solution to drift off the constraints that increases with the index. It is
usually only feasible for low index problems [LIU201593, Skvortsov12,
LIU2007748, Awawdeh09, Pryce98]. For references related to differential and
perturbation index see Hairer and Wanner [Hairer91], Campbell and Gear
[Campbell95], and Reid, Lin and Wittkopf [Reid01].

Note that the name dae misleadingly suggests that a dae can be partitioned
into differential equations and non-differential equations (algebraic
equations) where the latter are regarded as constraints. Consider the dae

(1)
$u^{\prime}u^{\prime\prime}+uu^{\prime}+x=0,((u^{\prime})^{2}+u^{2}+x^{2}-1)((u^{\prime})^{2}+u^{2}+x^{2}-4)=0$

where $u$ is a unknown function of $x$. Then
$((u^{\prime})^{2}+u^{2}+x^{2}-1)((u^{\prime})^{2}+u^{2}+x^{2}-4)=0$ is a
constraint even though it contains derivatives. Geometrically there are $2$
constraint components (spheres of radius $1$ and $2$).

In contrast to the above direct approach, indirect and widely used approaches
first use index reduction only by differentiation [Pryce01, Pryce98,
Pantelides88, Gear88, Fritzson14] followed by consistent initial point
determination [Brenan95, Taihei19, Pantelides88].

In this paper we make contributions to such indirect approaches. In particular
for polynomially nonlinear daes we apply a new efficient prolongation method
to reduce their index, which implicitly gives the hidden constraint components
of initial data, then determine consistent initial points using new methods
from real numerical algebraic geometry.

###  1.1 Previous Work

####  1.1.1 Consistent Point

Finding at least one consistent point on each constraint component of a dae,
is an important problem as it determines the initial point for a numerical
solution [Taihei19, Pantelides88]. Commonly used methods to obtain such
consistent initial points are the approximation method [Leimkuhler91] and the
transformation method [Vieira20011, Brown98]. The approximation method starts
with a guess for an initial point and then iteratively refines it
[Shampine02]. A good guess is critical for convergence of the iterative method
[Shampine02].

We note that most treatments assume there is just one constraint component,
and indeed that equations whose set of solutions correspond to this constraint
component (constraint equations) can be explicitly found. Geometrically
constraint components are projections from the higher dimensional space with
the derivatives regarded as indeterminates (the so-called Jet space of the
dae). For linear daes and polynomially nonlinear daes with rational
coefficients elimination algorithms are known for explicitly rendering
equations for their constraints. But no algorithms are known for the general
case of analytic daes, though there are some known for subclasses of analytic
daes.

####  1.1.2 Index Reduction

Indices are used to measure how far a dae is from a dae which includes its
missing constraints, or is implicitly equivalent to an ode. The Kronecker
index [Gear83, Lamour13] is applicable to constant coefficient linear dae. The
tractability index [Lamour13, Roswitha02, GUZEL06] and strangeness index
[KUNKEL94] applies to linear variable coefficient dae. Further, the
tractability index can be extended to a topological index [Tischendorf98] in
some applications, and the strangeness index also can be extended to non-
square dae [Kunkel06]. The perturbation index [Campbell95] is defined in terms
of perturbations of nonlinear autonomous dae. The differential index
[Campbell95, Campbell1995] is the minimum number of differentiation times
required to transform a dae system into an explicit ode system, and is used in
our paper.

In order to solve a dae accurately, index reduction is an essential and
important operation. The accurate numerical solution of a high index ($\geq
2$) dae is difficult to obtained directly [Shampine02]. Thus, we emphasize the
need to convert a higher index dae to a low ($\leq 1$) index dae. After
sufficient differentiation, all the time derivatives of the existing
differential variables can be replaced by new variables to realize the index
reduction [Pantelides88]. Gear [Gear88] proposed a method by repeatedly
finding algebraic equations and dealing with them by differential processing
until the system becomes an ode. However, these methods are notoriously hard
for large and non-linear systems. The arguments in [Pantelides88] and [Gear88]
depend on liberal use of the implicit function theorem for analytic functions
under tacitly assumed unstated rank conditions. The general finite termination
of prolongation of analytic systems of partial differential equations yielding
in finite steps involutive systems for which an existence and uniqueness
theorem can be given, was conjectured by Cartan. Kuranishi [Kuranishi57]
eventually proved this famous and difficult result that had eluded Cartan.

For polynomially nonlinear dae with rational coefficients, there are symbolic
differential-elimination algorithms that reduce index of dae, but these
algorithms are often unstable when applied to approximate dae and also have
high worst case complexity. Fortunately there are some efficient methods based
on bipartite graph preprocessing that can sometimes reduce the differential
index. These methods have been implemented in dae simulation packages such as
Dymola, Open-Modelica, MapleSim [Fritzson14], Simulink and Isamdae
[Caillaud20]. Such methods originated with work by Pantelides [Pantelides88]
who presented a graph-based preprocessing method that can sometimes by
prolongation reduce a dae to involutive (index 0 or 1) form containing the
underlying ode that decides consistent initial data for numerical solutions.
Crucially Pantelide’s method and its later developments have proven to be
successful often enough in applications that they have become a standard part
of the software environments mentioned above. Such developments include
Mattsson-Söderlind’s (MS) Method [Mattsson93] which employs an amending method
to introduce new variables to replace dummy derivatives [McKenzie17]. Pryce
[Pryce01] further generalized it to a more direct and widely applicable method
by solving an assignment problem. Zolfaghari, Taylor and Spiteri [ZOLF2021]
further extended Pryce method to the application of integro-
differential–algebraic equations.

####  1.1.3  Improved Structural Methods

Despite the success of structural analysis by index reduction, the methods may
fail for a dae if its Jacobian after differentiation is singular, and it is
essential to develop improved structure methods.

Campbell [Campbell93] proposed a direct method, which can regularize a dae in
theory by sufficiently differentiating the dae and simplifying it with an
elimination method. But the symbolic elimination process can be very complex
and inefficient for nonlinear dae.

Linear dae with constant coefficients can be transformed into the canonical
form of Weierstra$\beta$, the Kronecker index determined and then the dae can
be solved directly [Gerdts11]. This transformation is neither a strict
equivalence transformation [Iwata03] nor a unimodular transformation
[Murota95]. Murota [Murota95] proposed a general framework “combinatorial
relaxation” algorithm to compute the degree of a certain determinant based on
its upper bound, which is defined in terms of the maximum weight of a perfect
matching in an associated graph. Iwata [Iwata03] improved the combinatorial
relaxation algorithm by an equivalence transformation with constant matrices,
reducing computational complexity. X. Wu [Wu13] applied the modified
combinatorial relaxation algorithm to analyze the resulting error behavior. In
particular he gave an algorithm to detect and correct the error from
structural index reduction by matrix pencils.

Compared with the method of X. Wu et al. [Wu13], the LC-method of Tan et al.
[Tan17] also considers equations and their derivatives, with better results
for some nonlinear daes. Unfortunately, although this method may guarantee a
global equivalence transformation, it can only be used by its norm space. The
ES-method [Tan17] uses new variables to seek the solution in a projection of a
higher dimensional space, and it can be considered as a supplement for the LC-
method. The LC-method replaces equations, while the ES-method replaces
variables. If the global equivalence transformation of both methods or nether
of them is guaranteed, then the LC-method can be used, otherwise, the ES-
method can be used. Both of the above two methods can also be extended and
applied to some integro-differential–algebraic equations [ZOLF2021]. The
substitution method [Taihei19] aims to modify non-linear dae, and it is a
local equivalence method. Like the method of Campbell, it avoids excessive
elimination through targeted variable selection. For dae with high non-
linearity, this method is usually too complex to be applied. In order to avoid
the complexity of elimination calculations, the augmentation method [Taihei19]
adopts the principle similar to the ES-method, and is also a local equivalence
method.

###  1.2 Problem Statement

Mathematical models of circular motion in kinematics, mechanical structures
and chemical processes etc., often provide polynomially nonlinear daes.
Consequently they may have more than one solution component (see Example 1.2).
For global information about solutions, we need at least one consistent
initial point on each component. Such consistent initial points are hard to
obtain for a polynomial system with many variables by using symbolic
computation, e.g. by using Groebner Bases [Geddes1992] or Triangular
Decomposition [Collins1982]. Further, Newton iterative solvers usually require
a starting point sufficiently close to a solution. To obtain such global
information is one of the two main goals in this paper.

Moreover, the success of structural analysis methods for dae depends on the
regular Jacobian assumption after index reduction. In many cases, this
assumption is valid. However, we will present counterexamples from real
applications.

Such cases are called “degeneration” cases, which means the Jacobian matrix is
singular on a whole component, including two types: symbolic cancellation (see
Example 1.1) and numerical degeneration (see Example 1.2).

######  Remark 1.1.

In fact, singularities can occur only at special points along a solution. For
example in Equation (1), when $u$ is increasing, $u^{\prime}$ will gradually
change to $0$, leading to a singular Jacobian for the equation. We will not
consider this kind of problem in this paper.

######  Example 1.1.

Symbolic Cancellation: Consider the following dae [Taihei19]:

$\left\\{\begin{array}[]{rcl}\dot{x}_{1}+\dot{x}_{2}+{x}_{3}&=0\\\
\dot{x}_{1}+\dot{x}_{2}&=0\\\ {x}_{2}+\dot{x}_{3}&=0\end{array}\right.$

Symbolic cancellation occurs when the determinant of the Jacobian matrix of
the dae is identically zero. This case can be regularized by a number of
methods: a combinatorial relaxation method, a linear combination (LC) method,
and an expression substitution (ES) method [Tan17], a substitution method and
an augmentation method [Taihei19].

Unfortunately, there is little research on failure caused by numerical
degeneration. This could happen for a parametric dae model with a non-zero
determinant, where parameters take some specific values, and the determinant
equals zero after substituting any initial value on a component defined by the
constraints.

Figure 1: The Superposition Deformation of Beams under Collinear Bending
Moments

######  Example 1.2.

Numerical Degeneration:

In the bending deformation of a beam described in [Rans], the deformation of
any moment acting at a point $x$ satisfies ${\frac{\rm d^{2}}{{\rm
d}x^{2}}}y(x)=\frac{M(x)}{E(x)I(x)}$. When two collinear bending moments act
on the beam at the same time, the deformation is the superposition of the
effects of these moments. Suppose that the elastic deformation energy of the
bending moment at each point on the beam is linear in $\lambda$, then the
deformation variables $y_{1}(x)$, $y_{2}(x)$ satisfy:

$\left\\{\begin{array}[]{rcc}{\frac{\rm d^{2}}{{\rm
d}x^{2}}}y_{1}(x)+{\frac{\rm d^{2}}{{\rm
d}x^{2}}}y_{2}(x)+\frac{1}{5}\cdot(1-\sin(x))+y_{1}(x)&=&0\\\ \lambda\cdot
y_{1}(x)^{2}-y_{2}(x)^{2}&=&0\end{array}\right.$

In this example, the determinant of the Jacobian matrix is $-2(y_{2}+\lambda
y_{1})$. When the parameter $\lambda$ is equal to $1$, the constraint becomes
$y_{1}^{2}-y_{2}^{2}=(y_{1}+y_{2})(y_{1}-y_{2})=0$. In the view of physics,
this means that the elastic deformation energy of each point on the beam is
the same. Obviously, two consistent initial points can be selected from two
different components, respectively. If the point is on the component
$y_{1}-y_{2}=0$, then Pryce’s structural method works well. But for any
initial point on the component $y_{1}+y_{2}=0$, we always encounter a singular
Jacobian. Note that, this “bad” initial value usually can only be obtained
approximately, since there is no explicit expression for the roots of general
degree $5$ polynomials. Furthermore, such choices of the parameter values if
exist, often satisfy a nonlinear system and are only obtained numerically.
Therefore, we call this case numerical degeneration.

Such a degeneration is of potentially great importance in designing control
parameters in architecture, aviation and biochemistry. So a key question
deserving further study both theoretically and computationally which is
addressed in our paper is how to identify and solve such degenerated systems.

In summary, we aim to solve the following problems in this paper:

  * •

To detect at least one initial point on each real component of a polynomially
nonlinear dae.

  * •

To propose a global structural method to restore full rank Jacobian matrices
without algebraic elimination.

##  2 Preliminaries

In what follows we will use algorithmic aspects of the formal (Jet) theory of
differential equations [Reid01, Seiler2010]. Jet theory enables two inter-
related views of differential equations to be algorithmically and correctly
manipulated. One view is in terms of the maps as algebraic (non-differential)
equations, and the other view is in terms of local solutions of the
differential equations.

Let $\mathbb{I}$ be a nonempty sub-interval of $\mathbb{R}$. Let
$t\in\mathbb{I}\subset\mathbb{R}$ and suppose
$\bm{x},\bm{x}^{(1)},...,\bm{x}^{(\ell)}$ are vectors in $\mathbb{R}^{n}$,
where $\ell$ is a fixed positive integer. Here we consider maps
$\bm{F}:\mathbb{I}\times\mathbb{R}^{\ell n+n}\rightarrow\mathbb{R}^{m}$ which
are polynomially nonlinear in $\bm{x},\bm{x}^{(1)},...,\bm{x}^{(\ell)}$ and
real analytic in $t$, where possibly $m\not=n$.

An algebraic solution of this system is a point
$(t,\bm{x},\bm{x}^{(1)},...,\bm{x}^{(\ell)})\in\mathbb{I}\times\mathbb{R}^{\ell
n+n}$ that satisfies $\bm{F}(t,\bm{x},\bm{x}^{(1)},...,\bm{x}^{(\ell)})=0$. A
differential solution $\bm{x}(t)$ is a solution for $t$ in some subinterval of
$\mathbb{I}$ that satisfies the differential equations
$\bm{F}(t,\bm{x}(t),\bm{x}^{(1)}(t),...,\bm{x}^{(\ell)}(t))=0$ where now
$\bm{x}^{(k)}(t)$ denotes the $k$-order derivative of $\bm{x}(t)$. Sometimes
we will consider the system as an algebraic system and sometimes a
differential system, which should be clear from the context.

In particular, we consider systems which are not solved for their highest
derivatives, and regard such a system as a dae. The differential-algebraic
systems we consider have Jet form

(2) $\bm{F}(t,\bm{x},\bm{x}^{(1)},...,\bm{x}^{(\ell)})=0$

Let D be the formal total derivative operator with respect to independent
variable $t$:

(3) $\textbf{D}=\frac{\partial}{\partial
t}+\sum_{k=0}^{\infty}\bm{x}^{(k+1)}\frac{\partial}{\partial\bm{x}^{(k)}}$

Regarding $\bm{F}$ in its algebraic (jet) form a single prolongation of
$\bm{F}$ is the differentiation of each $F_{i}$ with respect to $t$, in which
$F_{i}$ is the $i$-th equation of $\bm{F}$, and it is denoted by

(4)
$\bm{F}^{(1)}=\textbf{D}\bm{F}\cup\textbf{D}^{0}\bm{F}=\\{\textbf{D}F_{1},...,\textbf{D}F_{n}\\}\cup\bm{F}$

It easily follows that the prolongation of $\bm{F}$ is a linear system with
respect to the “new” dependent variable $\bm{x}^{(\ell+1)}$. Thus, we can
rewrite

(5)
$\textbf{D}\bm{F}=\mathcal{S}(t,\bm{x},\bm{x}^{(1)},...,\bm{x}^{(\ell)})\cdot\bm{x}^{(\ell+1)}+G(t,\bm{x},\bm{x}^{(1)},...,\bm{x}^{(\ell)})$

where $\mathcal{S}$ is an $n\times n$ matrix called the “symbol matrix” and
$\bm{x}^{(\ell+1)}$ is a column vector and $G$ contains all the remaining
terms. Note that $\mathcal{S}$ is also the Jacobian matrix of $\bm{F}$ with
respect to its highest order derivative $\bm{x}^{(\ell+1)}$.

If we specify the prolongation order for $F_{i}$ to be $c_{i}$, then
$c_{i}\geq 0$, for $i=1,\dots,n$. For notational brevity, we will write
$(c_{1},...,c_{n})=\bm{c}\geq 0$. Then the prolongation of $\bm{F}$ up to the
order $\bm{c}$ is

(6)
$\bm{F}^{(\bm{c})}=\\{F_{1},\textbf{D}F_{1},...,\textbf{D}^{c_{1}}F_{1}\\}\cup\cdots\cup\\{F_{n},\textbf{D}F_{n},...,\textbf{D}^{c_{n}}F_{n}\\}=\textbf{D}^{\bm{c}}\bm{F}$

If $\bm{c}>\bm{0}$, then $\bm{F}^{(\bm{c})}$ also has linear structure similar
to (5). The number of equations of $\bm{F}^{(\bm{c})}$ is
$n+\sum_{i=1}^{n}c_{i}$.

###  2.1 Structural Prolongation Methods for DAE

In [Pryce01], Pryce reinterpreted Pantelides’ algorithm [Pantelides88] as an
assignment problem that reveals structural information about daes. This
structural method for square daes is a special case with roots in the work of
Jacobi [Ollivier09] and yields a local existence and uniqueness result. The
most important feature of Pryce’s method is that it finds all the local
constraints for a large class of square daes only using prolongation. A
generalization of this construction to partial differential-algebraic
equations can be found in [WRI09].

Suppose that the $k$-th order of derivative of $x_{j}$ occurs in $F_{i}$, then
the partial derivative $\partial F_{i}/\partial x_{j}^{(k)}$ is not
identically zero. The leading derivative of an equation or a system $\bm{F}$
with respect to $x_{j}$ is denoted by $\textsc{ld}(F,x_{j})$ and is the
highest order of derivative such that some $F_{i}\in\bm{F}$ depends on
$x_{j}^{(k)}$ for some $k\in\mathds{Z^{+}}$. We define the leading class
derivatives of a system $\bm{F}$ by

$\textsc{lcd}(\bm{F}):=\\{\textsc{ld}(\bm{F},x_{j}):1\leq j\leq n\\}\;$

Then we obtain an $n\times n$ matrix $\bm{\sigma}=(\sigma_{i,j})_{1\leq i\leq
n,1\leq j\leq n}$ which is called the signature matrix of $\bm{F}$ by Pryce
[Pryce01]:

(7) $(\sigma_{i,j})(\bm{F}):=\left\\{\begin{array}[]{ll}\hbox{ the order of
$\textsc{ld}(F_{i},x_{j})$;}\\\ -\infty,\;\;\hbox{otherwise.}\\\
\end{array}\right.$

Suppose that the highest order derivative of $x_{j}$ appearing in
$\bm{F}^{(\bm{c})}$, defined in Equation (6), is $d_{j}$. From the definition
of $\sigma_{i,j}$, clearly $d_{j}$ is the largest of $c_{i}+\sigma_{ij}$ for
$i=1,...,n$, which implies that

(8) $d_{j}-c_{i}\geq\sigma_{ij},\hbox{~{} for all~{}}i,j.$

Obviously, there are at most $n+\sum d_{j}$ derivatives and $n+\sum c_{i}$
equations in $\bm{F}^{(\bm{c})}$. The dimension of $\bm{F}^{(\bm{c})}$ usually
is $\sum d_{j}-\sum c_{i}$. Roughly speaking, finding all the constraints is
equivalent to minimizing the dimension of $\bm{F}^{(\bm{c})}$. This can be
formulated as an integer linear programming (ilp) problem in the variables
$\bm{c}=(c_{1},...,c_{n})$ and $\bm{d}=(d_{1},...,d_{n})$:

(9) $\delta(\bm{F})\left\|\begin{array}[]{l}\hbox{Minimize~{}~{}}\delta=\sum
d_{j}-\sum c_{i},\\\
\hbox{~{}~{}~{}~{}where~{}~{}}d_{j}-c_{i}\geq\sigma_{ij},\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}c_{i}\geq 0\\\ \end{array}\right.$

Let $\delta(\bm{F})$ be the optimal value of the problem (9).

######  Remark 2.1.

When a dae has no redundant equations, the optimal value $\delta(\bm{F})$ can
be regarded as degree of freedom (DOF) measure for the dae, and it also equals
the number of variables of $\bm{F}$ minus the number of equations of $\bm{F}$.
In this paper, we usually only consider cases without redundant equations in
theoretical derivation. Some cases with redundant equations will be addressed
in Section LABEL:ssec:sqare. We will also show the computational performance
of our approach in our experiments.

After we obtain the number of prolongation steps $c_{i}$ for each equation
$F_{i}$ by applying an ilp solver to Equation (9), we can construct the
partially prolonged system $\bm{F}^{(\bm{c})}$ using $\bm{c}$. We note that
$\bm{F}^{(\bm{c})}$ has a favorable block triangular structure enabling us to
compute consistent initial values more efficiently.

Without loss of generality, we assume $c_{1}\geq c_{2}\geq\cdots\geq c_{n}$,
and let $k_{c}=c_{1}$, which is closely related to the index of system
$\bm{F}$ (see [Pryce01]). The $r$-th order derivative of $F_{j}$ with respect
to $t$ is denoted by $F_{j}^{(r)}$. Then we can partition $\bm{F}^{(\bm{c})}$
into $k_{c}+1$ parts (see Table $1$), for $0\leq p\in\mathds{Z}\leq k_{c}$
given by

(10) $\bm{B}_{p}:=\\{F_{j}^{(p+c_{j}-k_{c})}:1\leq j\leq n,p+c_{j}-k_{c}\geq
0\\}.$

$\bm{B}_{0}$ | $\bm{B}_{1}$ | $\cdots$ | $\bm{B}_{k_{c}-1}$ | $\bm{B}_{k_{c}}$
---|---|---|---|---
$F^{(0)}_{1}$ | $F^{(1)}_{1}$ | $\cdots$ | $F^{(c_{1}-1)}_{1}$ | $F^{(c_{1})}_{1}$
| $F^{(0)}_{2}$ | $\cdots$ | $F^{(c_{2}-1)}_{2}$ | $F^{(c_{2})}_{2}$
|  | $\vdots$ | $\vdots$ | $\vdots$
|  | $F^{(0)}_{n}$ | $\cdots$ | $F^{(c_{n})}_{n}$
Table 1: The triangular block structure of $\bm{F}^{(\bm{c})}$ for the case of
$c_{p}=c_{p+1}+1$; For $0\leq p<k_{c}$, $\bm{B}_{i}$ has fewer jet variables
than $\bm{B}_{p+1}$.

Here, we call $\bm{B}_{k_{c}}$ the top block of $\bm{F}^{(\bm{c})}$ and
$\bm{F}^{(\bm{c}-1)}=\\{\bm{B}_{0},...,\bm{B}_{k_{c}-1}\\}$ the constraints.

Similarly, let $k_{d}=\max(d_{j})$ and we can partition all the variables into
$k_{d}+1$ parts:

(11) $\bm{X}^{(q)}:=\\{x_{j}^{(q+d_{j}-k_{d})}:1\leq j\leq n,q+d_{j}-k_{d}\geq
0\\}.$

For each $\bm{B}_{i},0\leq i\leq k_{c}$, we define the Jacobian Matrix

(12)
$\bm{\mathcal{J}}_{i}:=\left(\frac{\partial\bm{B}_{i}}{\partial\bm{X}^{(i+k_{d}-k_{c})}}\right).$

So $\bm{\mathcal{J}}_{k_{c}}$ is the Jacobian Matrix of the top block in the
table, and it is a square matrix.

######  Proposition 2.1.

Let $\\{\bm{\mathcal{J}}_{i}\\}$ be the set of Jacobian matrices of
$\\{\bm{B}_{i}\\}$. For any $0\leq i<j\leq k_{c}$, $\bm{\mathcal{J}}_{i}$ is a
sub-matrix of $\bm{\mathcal{J}}_{j}$. Moreover, if $\bm{\mathcal{J}}_{k_{c}}$
has full rank, then any $\bm{\mathcal{J}}_{i}$ also has full rank.

See [WRI09] for the proof.

Suppose $(t^{*},\bm{X}^{*})$ is a point satisfying the constraints
$\\{\bm{B}_{0},...,\bm{B}_{k_{c}-1}\\}$ and $\bm{\mathcal{J}}_{k_{c}}$ has
full rank at this point. Then Pryce’s structural method has successfully
finished the index reduction. However, it fails if $\bm{\mathcal{J}}_{k_{c}}$
is still singular, i.e. $\bm{\mathcal{J}}_{k_{c}}$ is degenerated.

Obviously, the definition of optimal value $\delta(\bm{F})$ is limited to
square systems, and we need to extend the definition for non-square systems
$\bm{F}^{(\bm{c})}$.

######  Definition 2.1.

Let a dae $\bm{F}$ consist of two blocks $\bm{A}$ and $\bm{B}$, where $\bm{F}$
contains $p$ equations and $n$ dependent variables $p\geq n$, and the
signature matrix of $\bm{A}$ be an $n\times n$ square matrix. So $\bm{B}$
contains the remaining $(p-n)$ equations. Let $\delta(\bm{A})$ be the optimal
value of the ilp of $\bm{A}$’s signature matrix. We define
$\delta(\bm{F})=\delta(\bm{A})-\\#eqns(\bm{B})$, where $\\#eqns(\bm{B})$ is
the number of equations in $\bm{B}$. Meanwhile, $\delta(\bm{F})$ also equals
the $DOF$ [Tan17] of $\bm{F}$, which equals the number of dependent variables
minus the number of equations in the prolongation of $\bm{F}$.

In the case of a square signature matrix of a dae $\bm{F}$, we have
$\\#eqns=0$, and the extended definition of $\delta(\bm{F})$ is equivalent to
the original definition.

######  Proposition 2.2.

Let $(\bm{c},\bm{d})$ be the optimal solution of Problem (9) for a given DAE
$\bm{F}$. Then $\delta(\bm{F})=\delta(\bm{F}^{(\bm{c})})=\sum d_{j}-\sum
c_{i}$.

Proof. For a prolonged dae system
$\bm{F}^{(\bm{c})}=\\{\bm{B}_{k_{c}},\bm{F}^{(\bm{c}-1)}\\}$, the signature
matrix of the top block $\bm{B}_{k_{c}}$ is square.

We construct a pair $(\hat{\bm{c}},\hat{\bm{d})}$, for $i=1,\cdots,n$ and
$j=1,\cdots,n$, $\hat{c}_{i}=0$ and $\hat{d}_{j}=d_{j}$. Since
$(\bm{c},\bm{d})$ is the optimal solution for $\bm{F}$, and $\bm{B}_{k_{c}}$
is the top block of $\bm{F}^{(\bm{c})}$, it follows that
$(\hat{\bm{c}},\hat{\bm{d})}$ is the optimal solution of $\bm{B}_{k_{c}}$,
$\delta(\bm{B}_{k_{c}})=\sum d_{j}$.

By Definition 2.1, and using $\\#eqns(\bm{F}^{(\bm{c}-1)})=\sum c_{i}$, we
obtain

$\delta(\bm{F}^{(\bm{c})})=\delta(\bm{B}_{k_{c}})-\\#eqns(\bm{F}^{(\bm{c}-1)})=\sum
d_{j}-\sum c_{i}=\delta(\bm{F})\,.$

$\square$

###  2.2 Framework for Improved Structural Methods

Many improved structural methods have been proposed to regularize the Jacobian
matrices of daes. See [Gerdts11, Iwata03, Murota95, Campbell93, Wu13] for
methods for linear dae. For non-linear daes, improved structural methods are
based on a combinatorial relaxation framework [Taihei19] with the following
steps:

Phase $1$.

Compute the solution ($\bm{c}$,$\bm{d}$) of ilp problem $\delta(\bm{F})$. If
there is no solution, the dae do not admit perfect matching, and the algorithm
ends with failure.

Phase $2$.

Determine whether $\bm{\mathcal{J}}_{k_{c}}$ is identically singular or not.
If not, the method returns $\bm{F}^{(\bm{c})}$ and halts.

Phase $3$.

Construct an new dae $\hat{\bm{F}}$, such that its solution space in $\bm{x}$
dimension is the same as dae $\bm{F}$ and
$0\leq\delta(\hat{\bm{F}})<\delta(\bm{F})$. Then go to Phase $1$.

######  Remark 2.2.

The key part of an improved structural method is to exploit different
regularization method for $\hat{\bm{F}}$ in Phase $3$. In this paper, our
global structural differentiation method mainly focuses on this phase.

Phase $2$ above is only to check for symbolic cancellation. As pointed out in
Example 1.2, $\det\bm{\mathcal{J}}_{k_{c}}$ may not be identically zero, but
$\det\bm{\mathcal{J}}_{k_{c}}=0$ at any consistent initial point of
$Z(\bm{F}^{(\bm{c})})$ — the zero set of $\bm{F}^{(\bm{c})}$. Since $\bm{F}$
is a polynomial system in $\\{\bm{x},\bm{x}^{(1)},...,\bm{x}^{(\ell)}\\}$,
$\bm{F}^{(\bm{c})}$ can be considered as a polynomial system in the variables
$\\{\bm{X}^{(0)},...,\bm{X}^{(k_{d})}\\}$. In the language of algebraic
geometry, it means that
$\det\bm{\mathcal{J}}_{k_{c}}\in\sqrt[\mathbb{R}]{\langle\bm{F}^{(\bm{c})}\rangle}$
or equivalently $Z_{\mathbb{R}}(\bm{F}^{(\bm{c})})\subseteq
Z_{\mathbb{R}}(\bm{\mathcal{J}}_{k_{c}})$.

In the rest of the paper, we usually suppress the subscript in
$\bm{\mathcal{J}}_{k_{c}}$ so it becomes $\bm{\mathcal{J}}$ unless the
subscript is needed.

######  Example 2.1.

Consider the following dae with dependent variables $x\left(t\right)$ and
$y\left(t\right)$:

(13) $\bm{F}=\\{2\,y{\frac{{\rm d^{2}}x}{{\rm d}t^{2}}}-x{\frac{{\rm
d^{2}}y}{{\rm d}t^{2}}}+2x\left({\frac{{\rm d}x}{{\rm
d}t}}\right)^{2}-{\frac{{\rm d}x}{{\rm d}t}}+\sin\left(t\right),y-x^{2}\\}.$

Applying the structural method yields $\bm{c}=(0,2)$ and $\bm{d}=(2,2)$. Then

$\bm{F}^{(\bm{c})}=[\\{2yx_{tt}-xy_{tt}+2x{x_{t}}^{2}-x_{t}+\sin(t),y_{tt}-2x_{t}^{2}-2xx_{tt}\\},\\{-2xx_{t}+y_{t}\\},\\{-x^{2}+y\\}]$

and the Jacobian matrix of the top block is
$\bm{\mathcal{J}}=\left(\begin{array}[]{cc}2y&-x\\\ -2x&1\\\
\end{array}\right)$.

Although the determinant of the Jacobian $2y-2x^{2}$ is not identically zero,
it must equal zero at any initial point, since the determinant belongs to the
polynomial ideal generated by the constraints, i.e.
$2y-2x^{2}\in\langle-x^{2}+y\rangle$.

Checking if a polynomial belongs to an ideal can be done by a standard ideal
membership test using a Gröbner basis of the ideal. In general, it is
challenging to compute the Jacobian determinant and the associated Gröbner
basis if the system is quite large. See the text [CoxLittleOshea07] for more
details about polynomial ideals, varieties and Gröbner bases. Algorithmic
algebraic geometry exploits Gröbner bases and related techniques to compute
features of solutions of general polynomial systems with exact (e.g. rational)
coefficients. Numerical versions of these algorithms, where exact numbers are
replaced with approximate numbers have largely been expensive and often
unstable.

In this paper, we propose a numerical approach based on real algebraic
geometry to detect such degenerated cases without using determinants or
Gröbner bases. It exploits a new generation of algorithms using a
fundamentally different and more thoroughly numerical approach, centered
around the concept of witness points on solution components and is discussed
in the next section.

###  2.3 Numerical Real Algebraic Geometry

Numerical algebraic geometry [SVW05, Hauenstein2017] was pioneered by Sommese,
Wampler, Verschelde and others (see [BHSW13, SommeseWampler05] for references
and background). The approach is built on witness points which arise by
slicing the complex variety with appropriate random planes of complementary
dimension. These complex witness points can be efficiently computed by
homotopy continuation solvers [Lee2008], and are theoretically guaranteed to
compute at least one such point on each solution component.

For the real case, the methods in [WuReid13, WRF17] yield real witness points
as critical points of the distance from a random hyperplane to the real
variety. Alternatively, the real witness points can be considered as critical
points of the distance from a random point to the real variety [Hauenstein12].

More precisely, to solve a polynomial system
$\bm{f}=\\{f_{1},...,f_{k}\\}\subset\mathbb{R}[x_{1},...,x_{n}]$, we first
choose a random point $\mathfrak{a}\in\mathbb{R}^{n}$, then there is at least
one point on each connected component of $V_{\mathbb{R}}(\bm{f})$ with minimal
distance to $\mathfrak{a}$ satisfying the following problem:

(14) $\displaystyle\min\;\sum_{i=1}^{n}(x_{i}-\mathfrak{a}_{i})^{2}/2$
$\displaystyle s.t.\hskip 28.45274pt\bm{f}(\bm{x})=0.$

This optimization problem can be formulated as a square system by using
Lagrange multipliers, i.e.

(15) $\bm{g}=\\{\bm{f},\sum_{i=1}^{k}\lambda_{i}\nabla
f_{i}+x_{i}-\mathfrak{a}_{i}\\}=0$

When $\bm{f}$ satisfies the regularity assumptions in [WuReid13], all the real
solutions of $\bm{g}=\bm{0}$ can be obtained by the homotopy continuation
method. These points are called real witness points of
$V_{\mathbb{R}}(\bm{f})$, where
$V_{\mathbb{R}}(\bm{f})=\\{\bm{x}\in\mathbb{R}^{n}:\bm{f}(\bm{x})=0\\}$. These
real solutions of the constraint equations provide initial points for every
component of a non-linear dae.

######  Definition 2.2.

For a polynomial system $\bm{f}$, let $\mathfrak{S}$ be the set of singular
points of $V_{\mathbb{R}}(\bm{f})$. If a finite set $W\subseteq\mathbb{R}^{n}$
contains at least one point on each connected component of
$V_{\mathbb{R}}(\bm{f})\backslash\mathfrak{S}$. Then this set is called the
real witness set of $V_{\mathbb{R}}(\bm{f})$ and these points are called real
witness points.

However, if $\bm{f}$ does not satisfy the regularity assumptions due to high
multiplicity or a non-real radical ideal, then we apply a critical point
approach [WuChenReid17] based on a penalty factor.

For example if $f=(x^{2}+y^{2}-1)^{2}$, then we choose a random point
$\mathfrak{a}=(a,b)$ and a large penalty factor $\beta$. The corresponding
regular system is

(16) $\left(\begin{array}[]{c}x\\\ y\\\ \end{array}\right)+\beta\cdot
f\cdot\left(\begin{array}[]{c}4x(x^{2}+y^{2}-1)\\\ 4y(x^{2}+y^{2}-1)\\\
\end{array}\right)=\left(\begin{array}[]{c}a\\\ b\\\ \end{array}\right)\\\ $

If the random point is $(a,b)=(1,1)$ and a large penalty factor $\beta=100000$
is chosen, then $4$ approximate solutions can be calculated by the homotopy
continuation method. Here, two of these solutions are far away from
components, which can be verified by substituting them into the equation $f$.
The verified solutions $[x=0.985220,y=0.172402],[x=-0.652031,y=-0.758442]$ are
close the real variety which is the unit circle. For more details, please see
[WuChenReid17].

##  3 Detecting Degeneration by Witness Points

To build a solid foundation of our theory, we need some results from the
theory of real analytic functions of several variables [KrantzParks02].

######  Definition 3.1.

A function $f$, with domain an open subset $U\subset\mathbb{R}^{n}$ and range
$\mathbb{R}$, is called real analytic on $U$, if for each $\bm{p}\in U$ the
function $f$ can be represented by a convergent power series in some
neighbourhood of $\bm{p}$.

######  Proposition 3.1 (Proposition 2.2.8 of [KrantzParks02]).

If $f_{1},...,f_{m}$ are real analytic in some neighbourhood of the point
$\bm{p}\in\mathbb{R}^{n}$ and $g$ is real analytic in some neighbourhood of
the point $(f_{1}(\bm{p}),...,f_{m}(\bm{p}))\in\mathbb{R}^{m}$, then the
composition of functions $g(f_{1}(\bm{x}),...,f_{m}(\bm{x}))$ is real analytic
in a neighborhood of $\bm{p}$.

######  Theorem 3.1 (Real Analytic Implicit Function Theorem [KrantzParks02]).

Given a set of equations
$f_{i}(x_{1},...,x_{m};y_{1},...,y_{n})=0,\;\;i=1,2,...,n$, where each $f_{i}$
is real analytic, suppose that
$(\bm{p},\bm{q})=(p_{1},...,p_{m};q_{1},...,q_{n})$ is a solution with
nonsingular Jacobian $\frac{\partial\bm{f}}{\partial\bm{y}}(\bm{p},\bm{q})$.

Then there exists a neighborhood $U\subset\mathbb{R}^{m}$ of $\bm{p}$ and a
set of real analytic functions $\phi_{j}:U\rightarrow\mathbb{R},j=1,2,...,n$,
such that $\phi_{j}(\bm{p})=q_{j},j=1,2,...,n$, and

$f_{i}(\bm{x};\phi_{1}(\bm{x}),...,\phi_{n}(\bm{x}))=0,i=l,2,...,n,$

hold for $\bm{x}\in U$.

######  Theorem 3.2 (Identity Theorem for Real Analytic Functions).

Given two real analytic functions $f$ and $g$ on an open and connected set
$U\subset\mathbb{R}^{n}$, if $f=g$ on a nonempty open subset $S\subseteq U$,
then $f=g$ on the whole set $U$.

######  Proof 3.3.

Define a set where $f$ and $g$ have the same power series:

$D=\left\\{\bm{c}\in
U:\frac{\partial^{|\bm{\mu}|}}{\partial\bm{x}^{\bm{\mu}}}f(\bm{c})=\frac{\partial^{|\bm{\mu}|}}{\partial\bm{x}^{\bm{\mu}}}g(\bm{c})\mbox{
for all }\bm{\mu}\right\\}$

where $\bm{\mu}$ is a multi-index in $(\mathbb{Z}^{+})^{n}$. Firstly, since
$f=g$ on a nonempty open subset $S$, we have $D\supseteq S$ and consequently
$D\neq\emptyset$. Secondly, $f$ and $g$ are real analytic on $U$, then for any
$\bm{c}\in D$ the power series have a non-zero radius of convergence. It
implies that $D$ is open. Meanwhile,

$D=\bigcap_{\mu}\left\\{\bm{c}\in
U:\frac{\partial^{|\bm{\mu}|}}{\partial\bm{x}^{\mu}}f(\bm{c})=\frac{\partial^{|\bm{\mu}|}}{\partial\bm{x}^{\bm{\mu}}}g(\bm{c})\right\\}$

which is an intersection of closed sets, so it is closed.

Therefore, by the connectedness of $U$, $D$ must be equal to the whole set
$U$. $\square$

######  Lemma 3.4.

Let $C$ be a connected real analytic manifold in $\mathbb{R}^{m+n}$ of
dimension $m$ and let $f$ be a real analytic function on $\mathbb{R}^{m+n}$.
Then the intersection $C\cap Z_{\mathbb{R}}(f)$ is equal to $C$ or has measure
zero over $C$.


# Geometric progressions in the sets of values of rational functions

Maciej Ulas

###### Abstract.

Let $a,Q\in\mathbb{Q}$ be given and consider the set
$\mathcal{G}(a,Q)=\\{aQ^{i}:\;i\in\mathbb{N}\\}$ of terms of geometric
progression with 0th term equal to $a$ and the quotient $Q$. Let
$f\in\mathbb{Q}(x,y)$ and $\mathcal{V}_{f}$ be the set of finite values of
$f$. We consider the problem of existence of $a,Q\in\mathbb{Q}$ such that
$\mathcal{G}(a,Q)\subset\mathcal{V}_{f}$. In the first part of the paper we
describe several classes of rational function for which our problem has a
positive solution. In particular, if $f(x,y)=\frac{f_{1}(x,y)}{f_{2}(x,y)}$,
where $f_{1},f_{2}\in\mathbb{Z}[x,y]$ are homogenous forms of degrees
$d_{1},d_{2}$ and $|d_{1}-d_{2}|=1$, we prove that
$\mathcal{G}(a,Q)\subset\mathcal{V}_{f}$ if and only if there are
$u,v\in\mathbb{Q}$ such that $a=f(u,v)$. In the second, experimental, part of
the paper we study the stated problem for the rational function
$f(x,y)=(y^{2}-x^{3})/x$. We relate the problem to the existence of rational
points on certain elliptic curves and present interesting numerical
observations which allow us to state several questions and conjectures.

###### Key words and phrases:

rational function, geometric progression, elliptic curve

###### 2000 Mathematics Subject Classification:

11D41

The research of the authors is supported by the grant of the National Science
Centre (NCN), Poland, no. UMO-2019/34/E/ST1/00094

Dedicated to Jaap Top on the occasion of his $60+2+\varepsilon$ birthday.

##  1\. Introduction and motivation

The problem of finding all integer solutions of the Diophantine equation

$x^{2}+7=2^{n}$

is a classical one. In fact, the above equation is the famous Ramanujan-Nagell
equation. Ramanujan conjectured that the equation has only five solutions
corresponding to $n=3,4,5,7,15$ [8]. This was proved by Nagell in [6]. Today,
by a Ramanujan-Nagell equation we understand a general equation of the form

$x^{2}+D=aQ^{n},$

where $a,Q,D\in\mathbb{Z}$ are fixed. The theory of such equations is quite
well developed and we have many explicit results concerning form of solutions
of particular equations or the number of integer solutions (see for example
[2, 3] and references given therein). It is clear that one can consider even
more general equation of the form

(1) $f(x)=aQ^{n},$

where $f$ is a given polynomial with rational coefficients and
$a,Q\in\mathbb{Z}$ are fixed. We know that if $f$ has at least two roots in
$\overline{\mathbb{Q}}$, then (1) has only finitely many solutions in integers
$x,n$. In fact, even if we allow $Q$ to vary, then, under mild conditions on
$f$, the equation (1) has only finitely many solutions in integers $x,Q,n$.
More on this line of research, or to be more precise, the realm of this type
of Diophantine equations, can be found in the monograph [9].

Let us observe that the result concerning the solvability of (1), where $a,Q$
are fixed, can be phrased in a different way. More precisely, let us consider
the set

$\mathcal{G}(a,Q)=\\{aQ^{i}:\;i\in\mathbb{N}\\},$

where $a,Q\in\mathbb{Q},aQ(Q^{2}-1)\neq 0$, i.e., an infinite set of values of
geometric progression with initial term equal to $a$ and the quotient equal to
$Q$, and the set $\mathcal{W}_{f}=\\{f(x):\;x\in\mathbb{Z}\\}$ \- the set of
values of the polynomial $f$. In other words, the equation (1) is solvable in
integers if and only if $\mathcal{G}(a,Q)\cap\mathcal{W}_{f}$ is non-empty.
From our discussion it follows that if $f\in\mathbb{Q}[x]$, then the only
possibility for $\mathcal{G}(a,Q)\cap\mathcal{W}_{f}$ to be infinite is when
$f$ is a power of a linear polynomial. Moreover, if $f$ is linear polynomial
then it is not difficult to construct infinitely many pairs $a,Q\in\mathbb{Z}$
such that the stronger condition $\mathcal{G}(a,Q)\subset\mathcal{W}_{f}$
holds. Indeed, let $f(x)=Ax+B$, where $A,B$ are fixed and observe that for
each $t,T\in\mathbb{Z}$ we have
$\mathcal{G}(f(t),AT+1)\subset\mathcal{W}_{f}$. This is a consequence of the
identity $f(v_{n})=f(t)(AT+1)^{n}$, where $v_{0}=f(t)$ and for $n\geq 1$ we
have $v_{n}=(AT+1)v_{n-1}+BT$.

Let us change a perspective and consider polynomials with more variables.
Then, it is possible that $\mathcal{G}(a,Q)$ is contained in the set of values
of $f$ even if the degree of $f$ is $\geq 2$. For example, if $p\equiv 1\mkern
4.0mu({\operator@font mod}\mkern 6.0mu4)$ is a prime number, then the set
$\mathcal{G}(1,p)=\\{p^{n}:\;n\in\mathbb{N}\\}$ is contained in the set of
values of the polynomial $x^{2}+y^{2}$. This simple observation motivated us
to consider, for a given $f\in\mathbb{Q}(x,y)$, the set

$\mathcal{V}_{f}=\\{f(u,v)\in\mathbb{Q}:\;u,v\in\mathbb{Q}\;\mbox{for
which}\;f(u,v)\;\mbox{determined}\\}$

and study the following general problem.

######  Problem 1.1.

Let $f\in\mathbb{Q}(x,y)$ be given. Does there exist rational numbers $a,Q$
such that $\mathcal{G}(a,Q)\subset\mathcal{V}_{f}$?

Let us note that the above problem in certain cases can be seen as a variant
of a well studied problem. More precisely, if $f\in\mathbb{Z}[x,y]$ is a
homogenous form of degree $d\geq 3$, then for given $a,Q\in\mathbb{Z}$ the
equation $f(x,y)=aQ^{i}$ is a special case of a so called Thue-Mahler
equation. It particular, it has only finitely many solutions $(x,y,i)$ in co-
prime integers $x,y$ and $i\in\mathbb{N}$. This was proved by Mahler in [5]
(generalizing an earlier result of Thue [10] concerning the equation
$f(x,y)=A$, where $A\in\mathbb{Z}\setminus\\{0\\}$ is fixed). However, we do
not assume that $x,y$ are integers. This assumption makes the problem more
interesting and, in certain cases, allows to use the techniques typically
applied in Diophantine problems where there is no real restriction between
rational and integral solutions (like in case of homogenous problems).

Let us describe the content of the paper in some details. In Section 2 we
present several classes of rational functions for which Problem 1.1 has a
solution. We also introduce a distinction between solutions of Problem 1.1 in
the case of weighted homogenous functions of degree $d$. In Section 3 we give
some observations concerning Problem 1.1 for the rational function
$f(x,y)=(y^{2}-x^{3})/x$ and shows connections with the existence of rational
points on certain elliptic curve (Theorem 3.1). As a consequence of our
investigations, we prove that for each $a\in\\{1,\ldots,10^{3}\\}$ such that
the elliptic curve $y^{2}=x^{3}+ax$ has positive rank, there are infinitely
many values of $Q\in\mathbb{Q}$ such that $Q$ is not a square and
$\mathcal{G}(a,Q)\subset\mathcal{V}_{f}$. Finally, in the last section we
consider Problem 1.1 for the rational function $f(x,y)=(y^{2}-x^{3})/x$ from a
computational point of view. We present results of various computer
experiments and formulate several questions, problems and conjectures which
may stimulate further research .

During our computational experiments we extensively used two computational
packages: the latest version of PARI/GP [7] (in case of rank computations) [7]
and Mathematica 13.2 (for symbolic computations and data manipulation) [13].

##  2\. Basic results

In this section we present certain classes of rational functions such that for
each element $f$ of an appropriate class there are values $a,Q\in\mathbb{Q}$
such that $\mathcal{G}(a,Q)\subset\mathcal{V}_{f}$.

######  Theorem 2.1.

Let $g_{1},g_{2},h_{1},h_{2}\in\mathbb{Q}(y)$ and suppose that $g_{1}h_{2}\neq
h_{1}g_{2}$ in $\mathbb{Q}(y)$. Let

$f(x,y)=\frac{xg_{1}(y)+g_{2}(y)}{xh_{1}(y)+h_{2}(y)}.$

Then, for any $a\in\mathbb{Q}\setminus\\{0\\}$ and $Q\neq\pm 1$ we have
$\mathcal{G}(a,Q)\subset\mathcal{V}_{f}$.

###### Proof.

Let $a,Q$ be given. For given $i\in\mathbb{N}$ we need to find rational
numbers $x_{i},y_{i}$ such that $f(x_{i},y_{i})=aQ^{i}$ for $i=0,1,\ldots$.
From our assumptions, for any $i\in\mathbb{N}$, we can chose
$y_{i}\in\mathbb{Q}$ such that $aQ^{i}h_{1}(y_{i})-g_{1}(y_{i})\neq 0$. Then,
it is enough to take

$x_{i}=\frac{g_{2}(y_{i})-aQ^{i}h_{2}(y_{i})}{aQ^{i}h_{1}(y_{i})-g_{1}(y_{i})}$

and note that $f(x_{i},y_{i})=aQ^{i}$ for $i\in\mathbb{N}$. ∎

######  Theorem 2.2.

Let $f(x,y)=ux^{2}+vxy+wy^{2}\in\mathbb{Q}[x,y]\setminus\\{0\\}$. There are
infinitely many pairs $a,Q\in\mathbb{Q}$ such that
$\mathcal{G}(a,Q)\subset\mathcal{V}_{f}$ is infinite.

###### Proof.

Multiplying $f$ by $4u$ we get that $4uf(x,y)=(2ux+vy)^{2}+(4uw-v^{2})y^{2}$.
We substitute $a=4ua^{\prime}$ and write $X=2ux+vy,Y=y,d=4uw-v^{2}$. Thus, the
problem of finding $a,Q\in\mathbb{Q}$ such that
$\mathcal{G}(a,Q)\subset\mathcal{V}_{f}$ is equivalent with the problem of
finding $a^{\prime},Q\in\mathbb{Q}$ such that
$\mathcal{G}(a^{\prime},Q)\subset\mathcal{V}_{F}$, where
$F(x,y)=x^{2}+dy^{2}$. To get the result we take
$a^{\prime}=r^{2}+ds^{2},Q=u^{2}+dv^{2}$, where $r,s,u,v\in\mathbb{Q}$ are
chosen in such a way that $rsuvF(r,s)F(u,v)\neq 0$. Then, noting that the form
$F$ is multiplicative, we get the result. ∎

######  Theorem 2.3.

Let $f(x,y)=\frac{f_{1}(x,y)}{f_{2}(x,y)}$, where
$f_{1},f_{2}\in\mathbb{Z}[x,y]$ are homogenous forms of degrees $d_{1},d_{2}$,
respectively. Let us assume that $|d_{1}-d_{2}|=1$. Then, for any
$Q\in\mathbb{Q}\setminus\\{-1,0,1\\}$ we have
$\mathcal{G}(a,Q)\subset\mathcal{V}_{f}$ if and only $a=f(u,v)$ for some
$u,v\in\mathbb{Q}$.

###### Proof.

Without loss of generality we can assume that $d_{1}-d_{2}=1$. If
$\mathcal{G}(a,Q)\subset\mathcal{V}_{f}$, then for some $u,v\in\mathbb{Q}$ we
have $a=f(u,v)$. To get the implication in the other side, we take
$x_{i}=uQ^{i},y_{i}=vQ^{i}$, where $u,v$ are rational parameters. Then we have

$f(x_{i},y_{i})=\frac{f_{1}(uQ^{i},vQ^{i})}{f_{2}(uQ^{i},vQ^{i})}=\frac{Q^{d_{1}i}f_{1}(u,v)}{Q^{d_{2}i}f_{2}(u,v)}=Q^{i}f(u,v).$

Thus, if $a=f(u,v)$ for some $u,v\in\mathbb{Q}$ such that $f(u,v)\neq 0$ we
have $f(x_{i},y_{i})=aQ^{i}$ for each $i\in\mathbb{N}$. Consequently
$\mathcal{G}(a,Q)\subset\mathcal{V}_{f}$. ∎

######  Remark 2.4.

Theorem 2.3 is a special (easily) provable part of the following general
observation. Suppose that $f\in\mathbb{Q}(x,y)$ is weighted homogeneous of
degree $d$, i.e., there is a pair $(w_{1},w_{2})$ of integers, called weights,
with such a property that for a variable $\lambda$, we have
$f(\lambda^{w_{1}}x,\lambda^{w_{2}}y)=\lambda^{d}f(x,y)$. Thus, to find
$a,Q\in\mathbb{Q}$ such that $\mathcal{G}(a,Q)\subset\mathcal{V}_{f}$ it is
enough to prove that we have
$\\{a,aQ,\ldots,aQ^{d-1}\\}\subset\mathcal{V}_{f}$. This is clear, if
$f(x_{i},y_{i})=aQ^{i}$ for $i=0,1,\ldots,d-1$, then
$f(x_{i}Q^{w_{1}j},y_{i}Q^{w_{2}j})=aQ^{i+dj}$ for $j\in\mathbb{Z}$ and thus
$\mathcal{G}(a,Q)\subset\mathcal{V}_{f}$.

Note that a rational function $f$ is homogenous of degree $d$ if and only if
is weighted homogenous of degree $d$ with weights $(w_{1},w_{2})=(1,1)$.

######  Example 2.5.

Let $f(x,y)=x^{3}+y^{3}$. We find that
$\mathcal{G}(13797,2)\subset\mathcal{V}_{f}$ which follows from observation
made in Remark 2.4 and the equalities

$f(-3,24)=13797,\quad f(-13,31)=2\cdot 13797,\quad f(-33,45)=2^{2}\cdot
13797.$

Let us also note that the problem of finding $a,Q$ such that
$\mathcal{G}(a,Q)\subset\mathcal{V}_{f}$ can be related to the Problem 1.1
with different rational function $f$. Indeed, let us recall that the curve
$C:\;x^{3}+y^{3}=A$ is birationally equivalent with the elliptic curve in
Weierstrass form $E:\;y^{2}=x^{3}-432A^{2}$. We have the maps

$\displaystyle\varphi:$
$\displaystyle\;C\ni(x,y)\mapsto\left(\frac{36A+y}{6x},\frac{36A-y}{6x}\right)\in
E,$ $\displaystyle\psi:$
$\displaystyle\;E\ni(x,y)\mapsto\left(\frac{12A}{x+y},\frac{36A(y-x)}{x+y}\right)\in
C.$

In other words, if $g(x,y)=y^{2}-x^{3}$, then we have

$\mathcal{G}(a,Q)\subset\mathcal{V}_{f}\Longleftrightarrow\mathcal{G}(-432a^{2},Q^{2})\subset\mathcal{V}_{g}.$

Let us consider $a=6,Q=7$, i.e., for $i=0,1,2$ we consider the curve
$E_{i}:\;y^{2}=x^{3}-432\cdot 6^{2}\cdot 7^{2i}$. One can check that the point
$S_{i}$ lies on the curve $E_{i}$ for $i=0,1,2$, where

$S_{0}=(28,80),\;S_{1}=(172,2080),\;S_{2}=(2353,113975).$

Then, we have $x_{i}^{3}+y_{i}^{3}=6\cdot 7^{i}$ for the corresponding points
$P_{i}=\psi(S_{i})=(x_{i},y_{i})$, where

$P_{0}=\left(\frac{37}{21},\frac{17}{21}\right),\;P_{1}=\left(\frac{449}{129},-\frac{71}{129}\right),\;P_{2}=\left(\frac{124559}{14118},-\frac{103391}{14118}\right).$

As a consequence we get the following

######  Corollary 2.6.

Let $\mathcal{C}=\\{x^{3}+y^{3}:\;x,y\in\mathbb{Z}\\}\subset\mathbb{Z}$. The
set $\mathcal{C}$ contains arbitrarily long geometric progressions.

###### Proof.

Let $n\in\mathbb{N}_{+}$ be given and $f(x,y)=x^{3}+y^{3}$. We proved that the
set $\mathcal{G}(6,7)\subset\mathcal{V}_{f}$. Let
$x_{i},y_{i}\in\mathbb{Q}\setminus\\{0\\}$ satisfy $f(x_{i},y_{i})=6\cdot
7^{i}$ for $i\in\mathbb{N}$. Let $D_{n}$ be a non-zero integer such that
$D_{n}\prod_{i=0}^{n}(x_{i}y_{i})\in\mathbb{Z}$. Then the numbers
$X_{i}=D_{n}x_{i},Y_{i}=D_{n}y_{i}$ are integers and we have

$f(X_{i},Y_{i})=6D_{n}^{3}7^{i}\quad\mbox{for}\quad i=0,1,\ldots,n,$

i.e., the set $\mathcal{C}$ contains geometric progression
$\\{6D_{n}^{3}7^{i}:\;i=0,\ldots,n\\}$ and the result follows. ∎

In the light of the proof of Corollary 2.6 on can ask the following

######  Question 2.7.

Do there exist integers $a,Q$ such that for each $i\in\mathbb{N}$ there is a
co-prime solution in integers $x,y$ of the Diophantine equation
$x^{3}+y^{3}=aQ^{i}$?

We should also note that in the case of $f\in\mathbb{Q}(x,y)$ which is
weighted homogenous of degree $d$ with weights $(w_{1},w_{2})$, the Problem
1.1 has a trivial solution. Indeed, for each $Q\in\mathbb{Q},Q(Q^{2}-1)\neq
0$, and any $u,v\in\mathbb{Q}$ such that $f(u,v)\neq 0$, we have
$\mathcal{G}(f(u,v),Q^{d})\subset\mathcal{V}_{f}$. This is simple consequence
of the identity

$f(uQ^{iw_{1}},vQ^{iw_{2}})=f(u,v)Q^{id}.$

Thus, in case of weighted homogenous rational function $f$ of degree $d$, we
need to consider various notions of solutions of Problem 1.1.

######  Definition 2.8.

Let $f\in\mathbb{Q}(x,y)$ be weighted homogenous of degree $d$. We say that
Problem 1.1 has

  1. (1)

a trivial solution $(a,Q)$ if and only if $Q$ is a $d$-th power of a rational
number and $\mathcal{G}(a,Q)\subset\mathcal{V}_{f}$;

  2. (2)

a non-trivial solution $(a,Q)$ of level $e$ if and only if $e|d,e<d,Q$ is
$e$-th power of a rational number, there is at least one prime $q$ such that
the $q$-adic valuation of $Q$ is equal to $e$ (or $-e$), and
$\mathcal{G}(a,Q)\subset\mathcal{V}_{f}$;

  3. (3)

a proper solution $(a,Q)$ if and only if $(a,Q)$ is a non-trivial solution of
level $1$.

We already observed that for each weighted homogenous rational function $f$,
Problem 1.1 has a trivial solution. On the other hand, for each $f$ from the
class of homogenous rational functions of degree 1, we know, via Theorem 2.3,
that Problem 1.1 has a proper solution. The same is true for the function
$f(x,y)=x^{3}+y^{3}$. In the next section we will investigate the existence
and properties of proper solutions of Problem 1.1 for
$f(x,y)=(y^{2}-x^{3})/x$. However, before we do that we note the following.

######  Theorem 2.9.

Let $d\in\mathbb{N}$ be odd.

  1. (1)

Let $f(x,y)=(y^{2}-x^{d})/x$. For each $Q\in\mathbb{Q}\setminus\\{-1\\}$ which
is not a square, there are infinitely many values of $a\in\mathbb{Z}$ such
that $(a,Q^{d-1})$ is a non-trivial solution of level $d-1$ of Problem 1.1.

  2. (2)

Let $f(x,y)=y^{2}-x^{d}$. For each $Q\in\mathbb{Q}\setminus\\{-1\\}$ which is
not a square, there are infinitely many values of $a\in\mathbb{Z}$ such that
$(a,Q^{d})$ is a non-trivial solution of level $d$ of Problem 1.1.

###### Proof.

We start with the case $f(x,y)=(y^{2}-x^{d})/x$. In this case $f$ is weighted
homogenous of degree $2(d-1)$ with weights $(2,d)$. Let us note that to get
the statement $\mathcal{G}(a,Q^{d-1})\subset\mathcal{V}_{f}$ it is enough to
prove that there exist rational solutions of the system of equations

$f(x_{0},y_{0})=a,\quad f(x_{1},y_{1})=aQ^{d-1}$

in variables $a,x_{i},y_{i},i=0,1$. Equivalently, by eliminating $a$, we
consider the equation

$f(x_{1},y_{1})=f(x_{0},y_{0})Q^{d-1}.$

It can be easily solved via the substitution $x_{i}=p_{i}T,y_{i}=q_{i}T^{m}$,
where $m=(d-1)/2,p_{i},q_{i}$ for $i=0,1$ are rational variables and we look
for an appropriate specialization of $T$. Indeed, after the substitution, our
equation reduces to the equation (in one variable $T$) of the form

$T^{d}[p_{0}p_{1}(p_{1}^{d-1}-p_{0}^{d-1}Q^{d-1})T+p_{1}q_{0}^{2}Q^{d-1}-p_{0}q_{1}^{2}]=0.$

Solving the equation defined by the second factor, we get an appropriate value
of $T$, together with the corresponding value of $a$ in the following form

$T=\frac{p_{0}q_{1}^{2}-p_{1}q_{0}^{2}Q^{d-1}}{p_{0}p_{1}(p_{1}^{d-1}-p_{0}^{d-1}Q^{d-1})},\quad
a=f(x_{0},y_{0})=f(p_{0}T,q_{0}T^{m})=\frac{T^{d-2}(q_{0}^{2}-p_{0}^{d}T)}{p_{0}}.$

As a consequence of our reasoning we see that by taking $X_{i}=x_{i\mkern
4.0mu({\operator@font mod}\mkern
6.0mu2)}Q^{2\lfloor\frac{i}{2}\rfloor},Y_{i}=x_{i\mkern 4.0mu({\operator@font
mod}\mkern 6.0mu2)}Q^{d\lfloor\frac{i}{2}\rfloor}$ we get
$f(X_{i},Y_{i})=aQ^{(d-1)i}$ for $i\in\mathbb{N}$, and thus
$\mathcal{G}(a,Q^{d-1})\subset\mathcal{V}_{f}$. If $Q$ is not a square then
our solution is non-trivial.

To get the statement for $f(x,y)=y^{2}-x^{d}$ we note that $f$ is weighted
homogenous of degree $2d$ with weights $(2,d)$. It is clear that to prove that
$\mathcal{G}(a,Q^{d})\subset\mathcal{V}_{f}$ it is enough to solve the system

$f(x_{0},y_{0})=a,\quad f(x_{1},y_{1})=aQ^{d}.$

or equivalently, the equation $f(x_{1},y_{1})=f(x_{0},y_{0})Q^{d}$. Using the
same substitutions and the same reasoning as in the case of the function
$(y^{2}-x^{d})/x$ we get the corresponding values of $T$ and $a$ in the
following form

$T=\frac{q_{0}^{2}-Q^{d}q_{1}^{2}}{p_{0}^{d}-Q^{d}p_{1}^{d}},\quad
a=f(x_{0},y_{0})=f(p_{0}T,q_{0}T^{m})=T^{d-1}(q_{0}^{2}-p_{0}^{d}T).$

∎

##  3\. Geometric progressions in the set $\mathcal{V}_{f}$ for
$f(x,y)=\frac{y^{2}-x^{3}}{x}$ and related elliptic curves

In the previous section we obtained some general result concerning Problem
1.1. Now we concentrate on a specific rational function. Let us put

$f(x,y)=\frac{y^{2}-x^{3}}{x}.$

We know that $f$ is weighted-homogenous of degree 4 with weights $(2,3)$ and
from the first part of Theorem 2.9, for each $Q\in\mathbb{Q}$ which is not a
square we have $\mathcal{G}(a,Q^{2})\subset\mathcal{V}_{f}$ non-trivially.
Before we will go on, let us note that without loss of generality one can
assume that $a,Q\in\mathbb{Z}$. Indeed, if $a=p/q,Q=u/v$, then by
multiplication of the equality $y^{2}=x^{3}+aQ^{i}x$ by $q^{6}v^{6i}$ we
easily get the equivalence

$\mathcal{G}(a,Q)\subset\mathcal{V}_{f}\;\Longleftrightarrow\;\mathcal{G}(pq^{3},uv^{3})\subset\mathcal{V}_{f}.$

Moreover, let us note that for any $t\in\mathbb{Q}$ we have: $(a,Q)$ is a
solution of Problem 1.1 if and only if $(at^{4},Q)$ is a solution of Problem
1.1.

Although the choice of $f$ may be seen arbitrary, it is not. This follows from
the fact that for any given $A\neq 0$ the curve $f(x,y)=A$ is an elliptic
curve, and the existence of $a,Q$ such that
$\mathcal{G}(a,Q)\subset\mathcal{V}_{f}$ is equivalent with the property that
for each $i\in\mathbb{N}$, the elliptic curve

$E_{i}(a,Q):\;y^{2}=x^{3}+aQ^{i}x$

contains rational point with $x\neq 0$. Actually, in the sequel we require
that the curve has a positive rank. This connection with elliptic curves
suggest several interesting questions.

It is not difficult to find an example of rational numbers $a,Q$ such that
$\mathcal{G}(a,Q)$ is infinite and $\mathcal{G}(a,Q)\subset\mathcal{V}_{f}$.
Indeed, let us take $Q=2$ and $a=47$. We thus consider elliptic curves
$E_{i}(47,2):\;y^{2}=x^{3}+47\cdot 2^{i}x$. One can compute the rank of
$E_{i}(47,2)$ and the set of generators of infinite order. We collect the
computed data in the table below.

$\begin{array}[]{|c|c|l|}\hline\cr i&r(E_{i}(47,2))&\mbox{generators
of}\;E_{i}(47,2)(\mathbb{Q})\\\ \hline\cr 0&1&(289/25,-5712/125)\\\
1&2&(2,14),(1504/81,65800/729)\\\ 2&1&(18,96)\\\
3&1&(4716544/18225,10271916928/2460375)\\\ \hline\cr\end{array}$

Table 1. Ranks and the generators for the elliptic curves
$E_{i}(47,2):\;y^{2}=x^{3}+47\cdot 2^{i}x,i=0,1,2,3$.

As a consequence we get that $\mathcal{G}(47,2)\subset\mathcal{V}_{f}$. In the
light of this result one can consider related problem. More precisely, let
$a\in\mathbb{Q}\setminus\\{0\\}$ be given and assume that the elliptic curve
$E:\;y^{2}=x^{3}+ax$ has a positive rank. Is it possible to find a
$Q\in\mathbb{Z}\setminus\\{-1,0,1\\}$ which is fourth power-free and the
elliptic curve $E_{i}(a,Q)$ has positive rank for $i=0,1,2,3$? We prove the
following:

######  Theorem 3.1.

Let $a\in\mathbb{Z}\setminus\\{0\\}$ and suppose that the elliptic curve
$E:\;y^{2}=x^{3}+ax$ has a positive rank. Let us consider the elliptic curve

$\mathcal{E}_{a}(p,v):\;Y^{2}=X(X-f_{1}(a,p,v))(X-f_{2}(a,p,v))=:F(X),$

where

$\displaystyle f_{1}$ $\displaystyle=ap^{3}(p^{2}+a)^{3}v^{4},$ $\displaystyle
f_{2}$ $\displaystyle=ap(p^{2}+a)((p^{2}+a)pv^{2}-1)((p^{2}+a)pv^{2}+1).$

If there are $p,v\in\mathbb{Q}$ such that $\mathcal{E}_{a}(p,v)$ has a
positive rank, then there are infinitely many values of fourth power-free
integers $Q$ which are not squares and such that
$\mathcal{G}(a,Q)\subset\mathcal{V}_{f}$.

###### Proof.

It is clear that we are interested in finding solutions of the following
system of Diophantine equations

(2)
$\frac{y_{1}^{2}-x_{1}^{3}}{ax_{1}}=Q,\quad\frac{y_{2}^{2}-x_{2}^{3}}{ax_{2}}=Q^{2},\quad\frac{y_{3}^{2}-x_{3}^{3}}{ax_{3}}=Q^{3}$

in variables $x_{i},y_{i},i=1,2,3$ and $Q$. To find a solution we make the
following substitution

(3) $x_{1}=v^{2},\;y_{1}=y,\quad x_{2}=pQ,\;y_{2}=qQ^{2},\quad
x_{3}=rQ^{2},\;y_{3}=sQ^{3}$

which reduces the system (2) to the system

(4) $Q=g_{1}(v,y)=g_{2}(p,q)=g_{3}(r,s),$

where

$g_{1}(v,y)=\frac{y^{2}-v^{6}}{av^{2}},\quad
g_{2}(p,q)=\frac{p^{3}+ap}{q^{2}},\quad g_{3}(r,s)=\frac{ar}{s^{2}-r^{3}}.$

We take $r=ap(p^{2}+a)$ and observe that the equation $g_{2}(p,q)=g_{3}(r,s)$
defines a quadric in variables $(q,s)$ which contains rational points at
infinity and can be parameterized in the following way

$q=-\frac{a^{3}p^{3}u^{2}-a^{3}-3a^{2}p^{2}-3ap^{4}-p^{6}}{2au},\quad
s=\frac{a^{3}p^{3}u^{2}+a^{3}+3a^{2}p^{2}+3ap^{4}+p^{6}}{2u}.$

With $q,r,s$ chosen in this way we left with one equation
$g_{1}(v,y)=g_{2}(p,q(p,u))$. If we treat $p,v$ as a rational independent
variables, then this equation defines the curve in the plane $(u,Y)$, over the
field $\mathbb{Q}(p,v)$, of the form

$\displaystyle\mathcal{C}_{a}:\;($ $\displaystyle
a^{3}p^{3}u^{2}-(p^{2}+a)^{3})^{2}y^{2}=$ $\displaystyle
a^{6}p^{6}v^{6}u^{4}-2a^{3}p(p^{2}+a)(p^{2}(p^{2}+a)^{2}v^{4}-2)v^{2}u^{2}+(p^{2}+a)^{6}v^{6}=:F_{a}(p,v,u).$

As a consequence of our reasoning, we see that each point $(u,Y)$ lying on the
curve $\mathcal{C}_{a}$, together with corresponding values of $q,r,s$ defined
above, lead to the value of $Q$ in the following form

$Q=Q_{a}(p,u)=\frac{4a^{2}p(p^{2}+a)u^{2}}{(a^{3}p^{3}u^{2}-(p^{2}+a)^{3})^{2}}.$

The genus of the curve $\mathcal{C}_{a}$ is equal to 1. Moreover,
$\mathcal{C}_{a}$ contains $\mathbb{Q}(p,v)$-rational point
$(0,(p^{2}+a)^{3}v^{3})$. Thus, the curve $\mathcal{C}_{a}$ is birationally
equivalent with a curve given by an appropriate Weierstrass equation. In fact,
$\mathcal{C}_{a}$ is birationally equivalent with the curve
$\mathcal{E}_{a}:\;Y^{2}=X(X-f_{1}(p,v))(X-f_{2}(p,v))$ from the statement of
our theorem, via an appropriate invertible map
$\varphi:\;\mathcal{C}_{a}\ni(u,Y)\mapsto(X,Y)\in\mathcal{E}_{a}$. We do not
present the form of the map because it is quite complicated (but can be easily
computed using standard method of finding birational model of a quartic curve
of the form $y^{2}=Au^{4}+Bu^{3}+Cu^{2}+Du+q^{2}$ with rational point $(0,q)$,
see [11, Theorem 2.17]).

Let us observe that the discriminant of $F_{a}$ with respect to $u$ is given
by

$D_{a}(p,v)=\operatorname{Disc}_{u}(F_{a}(p,v,u))=2^{12}a^{18}p^{10}(a+p^{2})^{10}v^{20}(p^{2}(p^{2}+a)^{2}v^{4}-1)^{2}.$

Because we are interested in values of $p=p_{0},v=v_{0}$ such that
$p_{0}(p_{0}^{2}+a)$ is not a square it is clear that $D_{a}(p_{0},v_{0})\neq
0$ and the corresponding curve $\mathcal{C}_{a}(p_{0},v_{0})$ is non-singular.
This is also true for the curve $\mathcal{E}_{a}(p_{0},v_{0})$.

If now $p_{0},v_{0}\in\mathbb{Q}$ are chosen in such a way, that
$f_{1}(p_{0},v_{0})f_{2}(p_{0},v_{0})\neq 0$ and the elliptic curve
$\mathcal{E}_{a}(p_{0},v_{0})$ obtained from $\mathcal{E}_{a}$ after
specialization $p=p_{0},v=v_{0}$, has infinitely many rational points, then we
get infinitely many rational values of $Q$. However, to finish the proof we
need to show that if $\mathcal{E}_{a}(p_{0},v_{0})$ has infinitely many
rational points, then we get infinitely many essentially different values of
$Q$. More precisely, we prove that there is an infinite set
$\mathcal{U}\subset\mathbb{Q}$ such that for each
$u^{\prime},u^{\prime\prime}\in\mathcal{U}$ the quotient
$Q_{a}(p_{0},u^{\prime})/Q_{a}(p_{0},u^{\prime\prime})$ is not a fourth power.
We will construct the sequence of finite sets
$(\mathcal{U}_{n})_{n\in\mathbb{N}}$ recursively in the following way. Let us
fix a point $P_{0}=(X_{0}^{\prime},Y_{0}^{\prime})$ of infinite order on
$\mathcal{E}_{a}(p_{0},v_{0})$ and write
$(u_{n},Y_{n})=\varphi^{-1}(nP_{0})\in\mathcal{C}_{a}(p_{0},v_{0})(\mathbb{Q})$.
Let us note that the point $(u,Y)\in\mathcal{C}_{a}(p_{0},v_{0})$ such that
$Q_{a}(p_{0},u)/Q_{a}(p_{0},u_{0})$ is a fourth power, lie on the variety

$\mathcal{W}_{a}(u_{0}):\;Q_{a}(p_{0},u)=Q_{a}(p_{0},u_{0})T^{4},\quad
Y^{2}=F_{a}(p_{0},v_{0},u),$

which, as an intersection of two surfaces, defines a curve over $\mathbb{Q}$.
One can check that the genus of $\mathcal{W}_{a}(u_{0})$ is equal to 5, and
thus there are only finitely many $u$’s which give a rational point. Thus,
there is a positive integer $N_{0}$ such that for all $N\geq N_{0}$ the
quotient $Q_{a}(p_{0},u_{n})/Q_{a}(p_{0},u_{0})$ is not a fourth power. We
thus take $\mathcal{U}_{0}=\\{u_{0}^{\prime},u_{1}^{\prime}\\}$, where
$u_{0}^{\prime}=u_{0},u_{1}^{\prime}=u_{N_{0}}$. Let us suppose that we
constructed the set
$U_{n-1}=\\{u_{0}^{\prime},u_{1}^{\prime},\ldots,u_{n}^{\prime}\\}\subset\\{u_{i}:\;i\in\mathbb{N}\\}$
with required properties, i.e., for each $i,j\in\\{0,\ldots,n\\},i\neq j$ the
quotient $Q_{a}(p_{0},v_{0},u_{i}^{\prime})/Q_{a}(p_{0},v_{0},u_{j}^{\prime})$
is not a fourth power. To find $u_{n+1}^{\prime}$ we note that for each
$i\in\\{0,\ldots,n\\}$ the curve $\mathcal{W}_{a}(u_{i})$ has only finitely
many rational solutions $(u,T,Y)$. Thus, there is an integer $N_{n+1}$ such
that for all $n\geq N_{n+1}$ there is no rational numbers $T,Y$ such that
$(u_{n},T,Y)$ lies on $\mathcal{W}_{a}(u_{i})$ for $i=0,\ldots,n$. As a
consequence of our reasoning we can take $u_{n+1}^{\prime}=u_{N_{n+1}}$ and
put $\mathcal{U}_{n+1}=\mathcal{U}_{n}\cup\\{u_{n+1}^{\prime}\\}$. Our
reasoning shows the existence of an increasing sequence
$(\mathcal{U}_{n})_{n\in\mathbb{N}}$ of sets with cardinality
$\\#\mathcal{U}_{n}=n+1$, each having a required property. To finish the
proof, it is enough to take
$\mathcal{U}=\lim_{n\rightarrow+\infty}\mathcal{U}_{n}$. ∎

It should be noted that if $p_{0},v_{0}\in\mathbb{Q}$ are such that
$\mathcal{E}_{a}(p_{0},v_{0})$ has positive rank, then the corresponding value
of $Q$ tends to be pretty large. To see this, let us consider the following
example.

######  Example 3.2.

Let us apply Theorem 3.1 for a specific value of $a$. We take $a=3$, which is
the smallest positive integer such that the curve $E:\;y^{2}=x^{3}+3x$ has a
positive rank. Next, we chose $p=4,v=2$ and the resulting elliptic curve is

$E_{3}(4,2):\;Y^{2}=X(X-21070848)(X-21070620).$

One can check that $E_{3}(4,2)$ has rank 1 and the generator of infinite part
is generated by the point $P=(X,Y)=(732246016/9,14683034857472/27)$. We have
that

$\varphi^{-1}(P)=(u,y)=\left(\frac{4204567}{4146944},-\frac{1592941018808}{199115595007}\right).$

Performing all necessary computations we found the value of $Q$ given by

$Q=Q_{3}\left(4,\frac{4204567}{4146944}\right)=19\cdot\left(\frac{476639376}{199115595007}\right)^{2}.$

Then we have the point $P_{i}$ of infinite order on $E_{i}(3,Q),i=0,1,2,3$,
where $P_{0}=(1,2)$ and

$\displaystyle P_{1}$
$\displaystyle=\left(4,\frac{1592941018808}{199115595007}\right),$
$\displaystyle P_{2}$
$\displaystyle=\left(\frac{17266067201278872576}{199115595007^{2}},\frac{78182031219520468152777449472}{199115595007^{3}}\right),$
$\displaystyle P_{3}$ $\displaystyle=\left(\frac{57\cdot
8633033600639436288^{2}}{39647020174991643330049^{2}},\frac{127652133024668050546\cdot
51798201603836617728^{2}}{39647020174991643330049^{3}}\right).$

######  Remark 3.3.

Let us note that the smallest positive integer $Q>1$ such that $(3,Q)$ is a
proper solution of Problem 1.1 is $Q=17$. The point $P_{i}$ is of infinite
order on $E_{i}(3,17),i=0,1,2,3$, where

$P_{0}=(1,1),\;P_{1}=(25,130),\;P_{2}=\left(\frac{121}{25},\frac{8206}{125}\right),\;P_{3}=\left(\frac{49}{121},\frac{102830}{1331}\right).$

Let

$\mathcal{R}=\\{a\in\mathbb{N}_{+}:\;\mbox{rank of the
curve}\;y^{2}=x^{3}+ax\;\mbox{is positive}\\}.$

We performed small computer search and found that for each
$a\in\mathcal{R}\cap[1,10^{3}]$, there are $p_{0},v_{0}\in\mathbb{N}$ such
that the curve $\mathcal{E}_{a}(p_{0},v_{0})$ has positive rank. In fact, in
the considered range of $a$, we can always take $p_{0}$ from the set
$\\{2,3,4\\}$ and $v_{0}$ form the set $\\{1,\ldots,10\\}$. Although small,
our computations suggest the following:

######  Conjecture 3.4.

For each $a\in\mathcal{R}$ there is a specialization $p=p_{0},v=v_{0}$ such
that $p_{0}(p_{0}^{2}+a)$ is not a square and the rank of the elliptic curve
$\mathcal{E}_{a}(p_{0},v_{0})$ is positive.

In light of Theorem 3.1 one can investigate the following related question.
Let $Q\in\mathbb{Z}\setminus\\{-1,0,1\\}$ be given. Is it possible to find an
$a\in\mathbb{Z}$ which is a fourth power-free and the elliptic curve
$E_{i}(a,Q)$ has positive rank for $i=0,1,2,3$? Equivalently, we are
interested in the solutions of the system od Diophantine equations

(5)
$a=f(x_{0},y_{0})=\frac{f(x_{1},y_{1})}{Q}=\frac{f(x_{2},y_{2})}{Q^{2}}=\frac{f(x_{3},y_{3})}{Q^{3}}.$

The common value of the fractions $f(x_{i},y_{i})/Q^{i},i=0,1,2,3$ is the
value of $a$ we are looking for. Let us put $x_{i}=p_{i}T,y_{i}=q_{i}T$ and
solve the equation $f(x_{i},y_{i})/Q^{i}=f(x_{i+1},y_{i+1})/Q^{i+1}$ with
respect to the variable $T$, for $i=0,1,2$. As a result, we see that to solve
(5), the value of $T$ need to be of the form

$T=\frac{p_{0}q_{1}^{2}-Qp_{1}q_{0}^{2}}{p_{0}p_{1}(p_{1}^{2}-Qp_{0}^{2})}=\frac{p_{1}q_{2}^{2}-Qp_{2}q_{1}^{2}}{p_{1}p_{2}(p_{2}^{2}-Qp_{1}^{2})}=\frac{p_{2}q_{3}^{2}-Qp_{3}q_{2}^{2}}{p_{2}p_{3}(p_{3}^{2}-Qp_{2}^{2})}.$

We arrived to the following conclusion: if $(x_{i},y_{i}),i=0,1,2,3$ solve the
system (5), then the corresponding values of $p_{i},q_{i},i=0,1,2,3,$ satisfy
the system

(6)
$\mathcal{C}^{\prime}_{Q}:\;\begin{cases}\begin{array}[]{lll}p_{1}p_{2}Q(p_{1}^{2}Q-p_{2}^{2})q_{0}^{2}+p_{0}p_{2}(p_{2}^{2}-p_{0}^{2}Q^{2})q_{1}^{2}+p_{0}p_{1}(p_{0}^{2}Q-p_{1}^{2})q_{2}^{2}&=&0,\\\
p_{2}p_{3}Q(p_{2}^{2}Q-p_{3}^{2})q_{1}^{2}+p_{1}p_{3}(p_{3}^{2}-p_{1}^{2}Q^{2})q_{2}^{2}+p_{1}p_{2}(p_{1}^{2}Q-p_{2}^{2})q_{3}^{2}&=&0,\end{array}\end{cases}$

and the corresponding value of $a$ has the form

$a=-\frac{(p_{1}^{3}q_{0}^{2}-p_{0}^{3}q_{1}^{2})(p_{1}q_{0}^{2}Q-p_{0}q_{1}^{2})}{p_{0}^{2}p_{1}^{2}(p_{1}^{2}-p_{0}^{2}Q)^{2}}.$

The variety $\mathcal{C}^{\prime}$ can be seen as a projective variety defined
over the function field $K=\mathbb{Q}(Q,p_{0},p_{1},p_{2},p_{3})$. This is
just an intersection of two quadratic surfaces, and thus, from geometric point
of view, we get a curve of genus 1. Unfortunately, it seems that there is no
general point on $\mathcal{C}^{\prime}$ and it is non-trivial problem to find
a specialization $\overline{p}=(p_{0},p_{1},p_{2},p_{3})$ such that the curve
$\mathcal{C}^{\prime}(\overline{p})$ has infinitely many rational points and
$(a,Q)$ is a proper solution of Problem 1.1.

######  Example 3.5.

Let $Q=2$. We know that $(47,2)$ is a proper solution of Problem 1.1 for
$f(x,y)=(y^{2}-x^{3})/x$. Using points from Table 1. one can take
$\overline{p}=\left(289/25,14,96,4716544/18225)\right)$ and note that the
curve $\mathcal{C}^{\prime}_{2}(\overline{p})$ contains the point
$S=(-5712/125,14,96,10271916928/2460375)$. Using this point as a point at
infinity, we get that $\mathcal{C}^{\prime}_{2}(\overline{p})$ is birational,
via an appropriate map $\chi$, with the elliptic curve with minimal
Weierstrass equation
$\mathcal{E}^{\prime}_{2}(\overline{p}):\;y^{2}=x^{3}+ax+b$, where

$\displaystyle a$ $\displaystyle=-114789376213793508149518054796962396588,$
$\displaystyle b$
$\displaystyle=-154452043874358875364810179220092982963254111141598049488.$

The curve $\mathcal{E}^{\prime}_{2}(\overline{p})$ contains the point of
infinite order

$\chi(S^{\prime})=\left(-\frac{32968008403251534251}{9},\frac{397625982502899147956961689375}{27}\right),$

where $S^{\prime}=(5712/125,14,96,10271916928/2460375)$. Thus, one can compute
the points $\chi^{-1}(m\chi(S^{\prime}))$ for $m\in\mathbb{N}_{+},m\neq 2,$
and get corresponding points on $\mathcal{C}^{\prime}_{2}(\overline{p})$, and
an appropriate value of $a_{m}(\overline{p},\overline{q})$. For $m=1$ we get
nothing new, i.e., $a_{1}(\overline{p},\overline{q})=47u^{4}$ for some
$u\in\mathbb{Q}$. For $m=3$, the value $a_{m}(\overline{p},\overline{q})$ is,
up to fourth power, an integer with 1564 digits.

As we will see in the next section, there are many $a$’s such that for given
$Q\in\\{2,\ldots,12\\}$ the pair is a proper solution of Problem 1.1.

##  4\. Computational observations, questions and conjectures

In this section we collect results of our computations and formulate some
questions and conjectures which may stimulate further research. To compute
ranks of several thousands of elliptic curves we used very fast procedure
ellrank implemented in PARI/GP [7]. The function ellrank attempts to compute
the rank of a given elliptic curve $E$. The function returns the vector
$[r,R,s,L]$. The rank of $E(\mathbb{Q})$ is between $r$ and $R$ (both
included), $s$ is related to the Tate-Shafarevitch group, and $L$ is a list of
independent, non-torsion rational points on the curve. To init the curve we
need to use procedure ellinit. Here is a sample session:

gp> E=ellinit([47,0]);
gp> ellrank(E)
[1, 1, 0, [[289/25, 5712/125]]]
So the rank of an elliptic curve $E:\;y^{2}=x^{3}+47x$ is one and an infinite
part of $E(\mathbb{Q})$ is generated by the point $(289/25,5712/125)$.
However, in some cases PARI/GP is not able to compute the rank exactly, like
in case of the elliptic curve $H:\;y^{2}=x^{3}+257x$:

gp> H=ellinit([257,0]);
gp> ellrank(H)
[0, 2, 0, []]
We get that the rank of $H(\mathbb{Q})$ is between 0 and 2. However, in such
cases we invoke the procedure ellanalyticrank which computes the pair $[r,d]$,
where $r$ is an analytic rank of an elliptic curve and $d$ the value of $d$-th
derivative at 1 of a $L$-function associated to $H$. We have

gp> analyticrank(H)
gp> [0, 7.4090738601929029389865884843731744616]
Thus, invoking Kolyvagin’s [4] together with Wiles’ [12] result on modularity
of elliptic curves over $\mathbb{Q}$, we know that the rank of $H$ is equal to
0.

After this digression, let us back to Problem 1.1 with
$f(x,y)=(y^{2}-x^{3})/x$. We performed additional computations and looked for
positive integers $a,Q$ such that $\mathcal{G}(a,Q)\subset\mathcal{V}_{f}$
with $Q\in\\{2,\ldots,12\\}$ and $a\leq 10^{5}$. For a given fourth power-free
$Q\in\mathbb{N}$ one can define the set

$C_{Q}:=\\{a\in\mathbb{N}:\;\mathcal{G}(a,Q)\subset\mathcal{V}_{f}\\}$

and its counting function

$C_{Q}(x)=\\#(C_{Q}\cap[1,x]).$

Because $(a,Q)$ is a solution of Problem 1.1 if and only if $(a,Q^{3})$ is a
solution, we get that for any square-free $Q$ we have equality of sets
$C_{Q}=C_{Q^{3}}$. Moreover, from the first part of Theorem 2.9 applied with
$d=3$, we know that $C_{Q^{2}}$ is infinite for each $Q$.

For $Q\in\\{1,\ldots,14\\}$ we have the following initial elements of the sets
$C_{Q}$ (we omit $C_{8}$ due to the equality $C_{8}=C_{2}$):

$\begin{array}[]{|c|l|}\hline\cr Q&C_{Q}\\\ \hline\cr
2&47,69,77,79,89,94,127,138,154,155,158,171,178,188,205,219,223,\ldots\\\
3&20,31,35,37,40,47,55,60,61,73,79,92,93,95,105,111,120,127,136,\ldots\\\
4&5,14,15,20,21,31,34,37,39,46,47,49,53,55,56,60,65,66,69,73,\ldots\\\
5&8,18,19,24,29,33,40,56,79,88,90,95,98,99,104,120,126,128,129,\ldots\\\
6&15,19,33,39,40,69,73,83,85,88,90,93,95,98,104,111,114,115,129,\ldots\\\
7&51,55,67,68,73,85,89,92,113,115,120,129,136,149,155,158,179,\ldots\\\
9&14,19,20,21,24,31,34,35,37,39,40,46,47,55,60,61,65,66,67,69,\ldots\\\
10&15,24,31,34,39,65,66,69,77,89,104,111,114,129,141,143,150,156,\ldots\\\
11&15,18,20,24,28,31,47,53,69,79,84,95,98,104,111,113,127,133,136,\ldots\\\
12&14,34,37,46,47,61,69,85,92,94,95,111,126,143,148,154,157,158,\ldots\\\
\hline\cr\end{array}$

Table 2. Initial elements of the sets $C_{Q}$ for $Q=2,\ldots,12,Q\neq 8$.

In the light of our computations one can ask the following

######  Question 4.1.

What is the order of magnitude of $C_{Q}(x)$?

Our numerical computations suggest that, at least for $Q\in\\{2,\ldots,12\\}$,
the function $C_{Q}(x)/x$ has a limit (see picture below), say $L(Q)$. The
conjectured estimations of $L(Q)$ can be easily read of from the values of
$C_{Q}(10^{5})$, which are as follows

$\begin{array}[]{|c|c|c|c|c|c|}\hline\cr Q&2&3&4&5&6\\\ \hline\cr
C_{Q}(10^{5})&13623&17766&37872&17130&17456\\\ \hline\cr\hline\cr
Q&7&9&10&11&12\\\ \hline\cr C_{Q}(10^{5})&16120&38562&13458&17682&17292\\\
\hline\cr\end{array}$

Table 3. Values of $C_{Q}(10^{5})$ for $Q=2,\ldots,12,Q\neq 8$.

In particular, $L(2)\approx 0.13623,L(3)\approx 0.17766,L(4)\approx 0.37872$
and so on.

Results of our computations strongly suggest the following:

######  Conjecture 4.2.

Let $f(x,y)=(y^{2}-x^{3})/x$.

  1. (1)

For each $Q\in\mathbb{N}_{\geq 2}$ the set $C_{Q}$ is not empty.

  2. (2)

For each $Q\in\mathbb{N}_{\geq 2}$ the set $C_{Q}$ is infinite.

######  Remark 4.3.

From the first part of Theorem 2.9 applied for $d=3$ we know that $C_{Q^{2}}$
is infinite. The method of proof can be used to get a lower bound for
$C_{Q^{2}}(x)$. Indeed, to make things explicit as much as possible, we take
in the proof of Theorem 2.9

$p_{0}=\frac{3}{4},\quad p_{1}=\frac{1}{4Q},\quad
q_{0}=3(3Q^{2}-1)(3Q^{2}+1)u,\quad q_{1}=(1-3Q^{2})(3Q^{2}+1)v.$

Then, the corresponding value of $a$ is given by

$a=a(u,v)=768Q^{3}(3Q^{2}-1)^{2}(3Q^{2}+1)^{2}(3Qu^{2}-v^{2})(3Q^{3}v^{2}-u^{2}).$

Now, if we fix $u\in\mathbb{N}$, then for each integer

$v\in\left(\frac{1}{Q\sqrt{3Q}}u,\sqrt{3Q}u\right)$

we have $a(u,v)>0$. It is possible that for some $u,v\in\mathbb{N}_{+}$ the
value $a(u,v)$ is a fourth power. However, the number of such possibilities is
finite because the (projective) curve $z^{4}=a(u,v)$ is of genus 3, and has
only finitely many primitive solutions. As a consequence we get that

$C_{Q^{2}}(x)>c(Q^{2})x^{1/4}$

for some computable constant $c(Q^{2})$. However, this bound is far weaker
than our expectations.

Figure 1. Plots of the functions $C_{Q}(x)$ for $Q=2,\ldots 6$ (left) and
$Q=7,\ldots,12,Q\neq 8$, (right) and $x\leq 10^{5}$. Figure 2. Plots of the
functions $C_{Q}(x)/x$ for $Q=2,\ldots 6$ (left) and $Q=7,\ldots,12,Q\neq 8$,
(right) and $x\leq 10^{5}$.

It is also quite interesting to note that

$\displaystyle\bigcap_{Q=2}^{12}C_{Q}\cap[1,10^{5}]=\\{$ $\displaystyle
32504,34023,36333,40399,41080,41415,55260,$ $\displaystyle
55965,73242,86415,99342\\},$

i.e., for each $Q\in\\{2,\ldots,12\\},i\in\mathbb{N}$ and each $a$ from the
above set, the elliptic curve $E_{i}(a,Q):\;y^{2}=x^{3}+aQ^{i}x$ has positive
rank. An additional check confirms that

$\displaystyle\bigcap_{Q=2}^{13}C_{Q}\cap[1,10^{5}]$
$\displaystyle=\\{32504,34023,36333,40399,41080,55260,73242,99342\\},$
$\displaystyle\bigcap_{Q=2}^{14}C_{Q}\cap[1,10^{5}]$
$\displaystyle=\\{40399,73242,99342\\}.$

However, the set $\bigcap_{Q=2}^{15}C_{Q}\cap[1,10^{5}]$ is empty. Our
computations suggest the following:

######  Question 4.4.

Let $N\in\mathbb{N}_{+}$ be given. Is the set $\bigcap_{Q=2}^{N}C_{Q}$ non-
empty?

We expect that the answer on the question above is YES. On the other hand, we
have the following:

######  Proposition 4.5.

We have $\bigcap_{Q=2}^{\infty}C_{Q}=\emptyset$.

###### Proof.

Let us suppose that the statement is not true and take
$a\in\bigcap_{Q=2}^{\infty}C_{Q}$. Without loss of generality we can assume
that $a$ is fourth power-free. Write $a=q_{1}q_{2}^{2}q_{3}^{3}$, where
$q_{1},q_{2},q_{3}$ are square-free pairwise co-prime integers. Let us take
$Q=q_{1}^{3}q_{2}^{2}q_{3}$ and observe that $a\not\in C_{Q}$. Indeed, we note
that the curve $E_{1}(a,Q):\;y^{2}=x^{3}+aQx$ is isomorphic over $\mathbb{Q}$
with the curve $E:\;y^{2}=x^{3}+x$. However, the rank of $E(\mathbb{Q})$ is
zero and the same is true for $E_{1}(a,Q)(\mathbb{Q})$ \- a contradiction. ∎

Next, we note that for given $Q_{1},Q_{2}\in\\{2,\ldots 11\\},Q_{1}\neq
Q_{2}$, our computations suggest that for sufficiently large $x$, the
difference $C_{Q_{1}}(x)-C_{Q_{2}}(x)$ has a constant sign, and in fact there
is $C_{Q_{1}}(x)-C_{Q_{2}}(x)\rightarrow\pm\infty$ with $x$ going to infinity.

In the light of our computations one can formulate the following general
problem.

######  Problem 4.6.

Let $\mathcal{Q}=\\{Q\in\mathbb{N}_{+}:\;Q\;\mbox{is fourth power free}\\}$.

  1. (1)

Characterize pairs $Q_{1},Q_{2}\in\mathcal{Q},Q_{1}>Q_{2}$ of positive
integers such that the sign of the function $C_{Q_{1}}(x)-C_{Q_{2}}(x)$ is
eventually constant.

  2. (2)

Does there exist $Q_{1},Q_{2}\in\mathcal{Q}$ such that
$C_{Q_{1}}(x)-C_{Q_{2}}(x)$ changes the sign infinitely often?

In the light of Theorem 3.1, for given $a\in\mathcal{R}$ one can define the
quantity

$m(a)=\operatorname{min}\\{Q\in\mathbb{N}_{\geq 2}:\;(a,Q)\;\mbox{is a proper
solution of Problem \ref{mainprob}}\\}.$

Although it is not clear that the function is well defined, results of our
computations suggest that this is the case. However, as we observed, the
corresponding value of $Q$ computed from rational point on the elliptic curve
$\mathcal{E}_{a}(p,v)$ constructed in Theorem 3.1 is much bigger than it is
expected. More precisely, from our computations it follows that for each
positive $a\in\mathcal{R}\cap[1,10^{4}]$, the value of $m(a)$ exists. The
values of $m(a)$ for $a\in\mathcal{R}\cap[3,60]$ are given in Table 4 below.

$\begin{array}[]{|c|ccccccccccccccc|}\hline\cr
a&3&5&8&9&13&14&15&18&19&20&21&24&28&29&31\\\ \hline\cr
m(a)&17&17&5&17&15&12&6&5&5&3&14&5&11&5&3\\\ \hline\cr\hline\cr
a&33&34&35&37&39&40&46&47&48&49&51&53&55&56&60\\\ \hline\cr
m(a)&5&10&3&3&6&3&12&2&17&17&7&11&3&5&3\\\ \hline\cr\end{array}$

Table 4. The values of $m(a)$ for $a\in\mathcal{R}\cap[3,60]$.

In fact, we have

$\\{m(a):\;a\in\mathcal{R}\cap[1,10^{4}]\\}=\\{2,3,5,6,7,10,11,12,13,14,15,17,21,22,23,26\\}.$

Let us write $\mathcal{R}=\\{a_{1},a_{2},\ldots\\}$, where $a_{i}<a_{i+1}$ for
$i\in\mathbb{N}_{+}$. On Figure 4 below, we present the plot of the function
$m(a_{n})$ for $n\in\\{1,\ldots,6005\\}$, i.e.,
$a_{n}\in\mathcal{R}\cap[1,10^{4}]$.

Figure 3. Plot of the function $f(n)=m(r_{n})$, where $n\leq 6005$.

In the table below, for each element $Q$ of the above set we present the
smallest value of $a\in\mathcal{R}$ such that $m(a)=Q$.

$\begin{array}[]{|c|cccccccc|}\hline\cr m(a)&2&3&5&6&7&10&11&12\\\ \hline\cr
a&47&20&8&15&51&34&28&14\\\ \hline\cr\hline\cr m(a)&13&14&15&17&21&22&23&26\\\
\hline\cr a&63&21&13&3&594&6295&2028&7255\\\ \hline\cr\end{array}$

Table 4. The smallest value of $a$ such that $m(a)=Q$, where
$Q\in\\{m(a):\;a\in\mathcal{R}\cap[1,10^{4}]\\}$.

We formulate the following:

######  Conjecture 4.7.

The function $m:\;\mathcal{R}\rightarrow\\{Q\in\mathbb{N}_{+}:\;Q\;\mbox{is
fourth power-free}\\}$ is well defined.

It is clear that Conjecture 3.4 implies Conjecture 4.7. However, it is not
clear whether there is an implication in the other side.

######  Conjecture 4.8.

The function $m:\;\mathcal{R}\rightarrow\\{Q\in\mathbb{N}_{+}:\;Q\;\mbox{is
fourth power-free}\\}$ is onto and each element in its image is a limit point.
In particular

$\liminf_{a\in\mathcal{R}}m(a)=2,\quad\limsup_{a\in\mathcal{R}}m(a)=+\infty.$

One can also ask about average behaviour of the function $m(a_{n})$. On Figure
5 we plot the function

$\mathcal{A}(x)=\frac{\sum_{i\leq x}m(a_{i})}{x}$

for $x\leq 6005$. Based on our computations we dare to formulate the
following:

######  Conjecture 4.9.

If Conjecture 4.7 is true, then the function $\mathcal{A}$ is bounded.

Figure 4. Plot of the function $\mathcal{A}(x)$, where $x\leq 6005$.

In the light of our findings concerning the questions around Problem 1.1 for
$f(x,y)=(y^{2}-x^{3})/x$, one can ask and investigated similar questions in
the case of the polynomial $f(x,y)=y^{2}-x^{3}$, both form computational and
theoretical point of view. To be more precise, we are interested in the
existence of proper solutions of Problem 1.1, i.e., $Q$ is neither as quare
nor a cube and for each $i\in\mathbb{N}$ the rank of elliptic curve

$E^{\prime}_{i}(a,Q):\;y^{2}=x^{3}+aQ^{i}$

is positive. We show that $(3,2)$ is a proper solution of Problem 1.1. To see
this, one can compute the following data:

$\begin{array}[]{|c|c|l|}\hline\cr i&r(E^{\prime}_{i}(3,2))&\mbox{generators
of}\;E^{\prime}_{i}(3,2)(\mathbb{Q})\\\ \hline\cr 0&1&(-2,5)\\\
1&1&(1/4,65/8)\\\ 2&1&(4,14)\\\ 3&1&(-2,-16)\\\ 4&2&(16,68),(-8,4)\\\
5&1&(48217/5041,-15728083/357911)\\\ \hline\cr\end{array}$

Table 5. Ranks and the generators for the elliptic curves
$E_{i}^{\prime}(3,2):\;y^{2}=x^{3}+3\cdot 2^{i},i=0,\ldots,5$.

Consequently, the pair $(32,2)$ is a proper solution of Problem 1.1 for
$f(x,y)=y^{2}-x^{3}$.

We formulate the following:

######  Problem 4.10.

Investigate properties of the set of proper solutions of Problem 1.1 for
$f(x,y)=y^{2}-x^{3}$. In particular prove the following:

  1. (1)

For each $a\in\mathbb{Z}$ such that the elliptic curve $y^{2}=x^{3}+a$ has
positive rank, there are infinitely many values of $Q\in\mathbb{Z}$ such that
$(a,Q)$ is a proper solution of Problem 1.1.

  2. (2)

For each $Q\in\mathbb{Z}\setminus\\{-1,0,1\\}$, there are infinitely many
values of sixth power-free integers $a$ such that $(a,Q)$ is a proper solution
of Problem 1.1.

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