# Sufficient conditions for the surjectivity of radical curve parametrizations

Jorge Caravantes1, J. Rafael Sendra2, David Sevilla3 and Carlos Villarino4
1 2 4 Universidad de Alcalá
Dpto. de Física y Matemáticas
28871 Alcalá de Henares (Madrid), Spain
3 Centro U. de Mérida
Universidad de Extremadura
Av. Santa Teresa de Jornet 38, 06800 Mérida (Badajoz), Spain
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###### Abstract

In this paper, we introduce the notion of surjective radical parametrization
and we prove sufficient conditions for a radical curve parametrization to be
surjective.

Keywords. Algebraic curve, radical parametrization, surjective
parametrization, missing points.

##  1 Introduction

Let $\mathcal{V}$ be a curve or an algebraic surface over $\mathbb{C}$ for
which a parametrization $\mathcal{P}$ is available. This parametrization can
be visualized as a map from a subset of the parameter space on $\mathcal{V}$.
It therefore makes sense to talk about the surjectivity of $\mathcal{P}$.

If the parametrization is not surjective, when working with it instead of the
implicit equations of $\mathcal{V}$, information about the part of
$\mathcal{V}$ not covered by $\mathcal{P}$ is lost. This fact may have
undesirable consequences in some applications since the information sought in
$\mathcal{V}$ may lie out of the image of $\mathcal{P}$; in the introduction
of [9] this phenomenon is illustrated with some examples. Thus, the need to
study the surjectivity of parametrizations arises.

If $\mathcal{V}$ is a curve and $\mathcal{P}$ is a rational parametrization,
the problem is essentially solved in [6] (see [1] and [2] for different
approaches) where, in addition to giving sufficient conditions to guarantee
surjectivity, it is shown that, at most, only one point can be missed via
$\mathcal{P}$, and it is shown how to detect it. In addition, it is also
proved that every rational curve can be surjectively parametrized.

When $\mathcal{V}$ is a surface, even if $\mathcal{P}$ is a rational
parametrization, the problem is considerably more complicated. In fact, in [4]
it is shown that there are rational surfaces that do not admit surjective
rational parametrizations. One way to overcome this difficulty is to look for
finite families of parametrizations such that the union of their images covers
$\mathcal{V}$ (see [5], [10], [9]). On the other hand, in [7] particular
families of surfaces are presented that do admit surjective rational
parametrizations.

In this paper, we apparently reduce the complexity of the problem by working
with curves instead of surfaces but increase the difficulty by allowing the
parametrization to be radical; a radical parametrization is, intuitively, a
tuple of rational functions whose numerators and denominators are nested
expressions of radicals of polynomials with Jacobian rank equal to one, see
[8] for more details. Therefore, this work should be considered as a first
step towards the general study of surjectivity in the radical case, leaving
open problems for future work such as obtaining, if feasible, surjective
radical parametrizations. This article focuses mainly on obtaining sufficient
conditions to guarantee surjectivity.

The first obstacle we face is to adequately define what we mean by
surjectivity in the radical case. Let us see the problem in a trivial example.
The rational parametrization
$\left(\frac{2t}{t^{2}+1},\frac{t^{2}-1}{t^{2}+1}\right)$ of the unit circle
covers all the curve except the north pole point $(0,1)$. However, if we
consider the radical parametrization $(t,\sqrt{1-t^{2}})$ of the same curve,
taking the square root as a function (i.e. just choosing one of the two
possibilities for every radicand), we can only aspire to cover one half of the
curve. Our approach is to refer to the union of the images of the given
parametrization and all its conjugates. Thus, in the example above, we would
be simultaneously talking about $(t,\pm\sqrt{1-t^{2}})$ and, now, the union of
the images is the entire circle; see Example 2.4 and Remark 3.13.

Since, in the case of rational curve parametrizations, the only point of the
curve that can be lost is the image of infinity, a sufficient condition for
surjectivity is that at least one of the rational functions of the
parametrization has a numerator with a degree strictly bigger than the degree
of the denominator. With the intention of imitating this result in the radical
case, we introduce weights to adequately define the degree of polynomials with
nested radicals, see Def. 2.6. However, this does not solve the problem
completely (see Example 2.7). The reason is that indeterminations of type 0/0
can appear. To overcome this difficulty we introduce the notion of guilty
polynomial (see Def. 3.5), and the weaker notion of suspicious polynomial (see
Def. 4.9), to finally give a sufficient condition of radical surjectivity, see
Theorem 3.8 and Corollary 4.13.

The paper is structured as follows: in Section 2 we recall some notions and
results of radical parametrizations, we introduce the notion of surjectivity
for a radical parametrization, and we outline our strategy. In Section 3 we
introduce the notion of guiltiness and we prove the main Theorem 3.8. Some
consequences are derived there. In Section 4 we discuss the hypotheses of the
main theorem and present an alternative condition to guiltiness, namely being
suspicious, that is more convenient from a computational point of view but in
principle more restrictive (see Theorem 4.12). In Section 5 we conclude the
article with bounds on the number of missing points in the case when some of
our conditions for surjectivity do not hold.

##  2 Notation and preliminaries

A radical parametrization is a tuple $(x_{1},\ldots,x_{n})$ of elements of the
last field of a radical tower
$\mathbb{F}_{0}\subseteq\cdots\subseteq\mathbb{F}_{m}$ where
$\mathbb{F}_{0}=\mathbb{C}(t)$ and
$\mathbb{F}_{i}=\mathbb{F}_{i-1}(\delta_{i})$ such that
$\delta_{i}^{e_{i}}\in\mathbb{F}_{i-1}$ (see more details in [8]). We will
work with the polynomial ring $\mathbb{C}[t,\Delta_{1},\ldots,\Delta_{m}]$
where weights on the variables will be introduced later. In this ring lie the
defining polynomials of the $\delta_{i}$, namely
$E_{i}:=\Delta_{i}^{e_{i}}-g_{i}(t,\Delta_{1},\ldots,\Delta_{i-1})$ where
$\deg_{\Delta_{j}}(g_{i})<e_{j}$ for all $j=1,...,i-1$. We will denote the
$\Delta_{i}$ collectively as $\,\overline{\Delta}\,$, and similarly for other
indexed names.

######  Remark 2.1.

Any rational function in $\mathbb{C}(t,\,\overline{\delta}\,)$ can be written
as a polynomial in $\mathbb{C}(t)[\,\overline{\delta}\,]$.

Recall from [8] that a radical parametrization of a curve, among other stuff,
incorporates a pair $(\mathcal{P},\,\overline{E}\,)$, where

${\mathcal{P}(t,\,\overline{\Delta}\,)=\left(\displaystyle\frac{p_{1}(t,\,\overline{\Delta}\,)}{q_{1}(t,\,\overline{\Delta}\,)},\ldots,\frac{p_{n}(t,\,\overline{\Delta}\,)}{q_{n}(t,\,\overline{\Delta}\,)}\right)}$
(1)

is an element of $\mathbb{C}(t,\,\overline{\Delta}\,)^{n}$, and
$\,\overline{E}\,=(E_{1},\ldots,E_{m})$. We will use throughout this paper the
following definition of surjectivity:

######  Definition 2.2.

We say that the parametrization is surjective or normal when the set

$X(\mathcal{P}):=\left\\{{\overline{x}\in\mathbb{C}^{n}}\left|\
\exists\,(t,\,\overline{\Delta}\,)\in\mathbb{C}^{m+1}\mbox{ \rm s.t.
}\begin{array}[]{l}E_{i}(t,\,\overline{\Delta}\,)=0,\
\forall\,i=1,\ldots,m;\cr\mathcal{P}(t,\,\overline{\Delta}\,)\mbox{ \rm is
well defined;
and}\cr\mathcal{P}(t,\,\overline{\Delta}\,)={\overline{x}}\end{array}\right.\right\\}$

is Zariski closed in $\mathbb{C}^{n}$.

Let us call $\mathcal{D}_{\mathcal{P}}$ the algebraic subset of
$\mathbb{C}^{m+n+2}$ of points $(t,\,\overline{\Delta}\,,\overline{x},z)$
satisfying the equations $E_{i}(t,\,\overline{\Delta}\,)=0$, $i=1,\ldots,m$;
$p_{i}(t,\,\overline{\Delta}\,)-x_{i}\,q_{i}(t,\,\overline{\Delta}\,)=0$;
$i=1,\ldots,n$; and
$z\cdot\mathrm{lcm}(q_{1}(t,\,\overline{\Delta}\,),\ldots,q_{n}(t,\,\overline{\Delta}\,))-1=0$.
We have then the projection

$\begin{array}[]{ccc}\mathcal{D}_{\mathcal{P}}&{\longrightarrow}&\overline{X(\mathcal{P})}{\subset\mathbb{C}^{n}}\\\
{(t,\,\overline{\Delta}\,,\overline{x},z)}&\mapsto&{\overline{x}}\end{array}$
(2)

where $\overline{X(\mathcal{P})}$ denotes the Zariski closure of
$X(\mathcal{P})$. Note that every constructible set involved is unidimensional
(see [8]). Then, the set of missing points
$\overline{X(\mathcal{P})}\setminus{X(\mathcal{P})}$ is finite. Observe that
the radical variety defined in [8], in our case, radical curve, is an
irreducible component of $\overline{X(\mathcal{P})}$, so, when irreducible,
$\overline{X(\mathcal{P})}$ is the whole radical variety.

######  Remark 2.3.

The definition of surjective in Def. 2.2 corresponds to the usual notion of
surjective map in the following way. For each combination of possible branches
in (1), we can introduce a map from a subset of $\mathbb{C}$ to
$\mathbb{C}^{n}$. Surjectivity in Def. 2.2 requires that
$\overline{X(\mathcal{P})}=X(\mathcal{P})$ (see (2)), and hence that the union
of the images of all these maps is $\overline{X(\mathcal{P})}$.

######  Example 2.4.

Let $\mathcal{P}(t,\Delta)=(t,\Delta)$ where $E:=\Delta^{2}-(1-t^{2})$. Then,
$\mathcal{D}_{\mathcal{P}}$ is

$\mathcal{D}_{\mathcal{P}}=\left\\{(t,\Delta,x,y,z)\in\mathbb{C}^{5}\,\left|\,\begin{array}[]{r}x-t=0\\\
y-\Delta=0\\\ z=1\\\ \Delta^{2}-(1-t^{2})=0\end{array}\right.\right\\}.$

By the Closure Theorem (see [3, Ch. 3.2, Th. 3 or Ch. 4.4, Th. 3]) we have
that $\overline{X(\mathcal{P})}$ is the curve
$\mathcal{C}:=\\{(x,y)\in\mathbb{C}^{2}\,|\,x^{2}+y^{2}=1\\}$. Let us now
analyze $X(\mathcal{P})$, that is

$X(\mathcal{P}):=\left\\{(x,y)\in\mathbb{C}^{2}\left|\
\exists\,(t,\Delta)\in\mathbb{C}^{2}\quad\mathrm{s.t.}\begin{array}[]{l}\Delta^{2}=1-t^{2}\cr(t,\Delta)=(x,y).\end{array}\right.\right\\}$

First, we observe that $X(\mathcal{P})\subset\mathcal{C}$ since
$t^{2}+\Delta^{2}=1$. On the other hand, if $(a,b)\in\mathcal{C}$, taking
$t=a$, and $\Delta$ as one of the roots of $\sqrt{1-a^{2}}$, we get that that
$(a\,b)\in\mathcal{D}_{\mathcal{P}}$, and hence
$X(\mathcal{P})=\overline{X(\mathcal{P})}$. So $X(\mathcal{P})$ is Zariski
closed and hence, by Def. 2.2, $\mathcal{P}$ is surjective.

On the hand $\mathcal{P}$ provides two map, namely, $(t,\pm\sqrt{1-t^{2}})$,
and the union of their images is $\mathcal{C}$.

Recall the notation $V(I)$ for the algebraic subset of the affine space
defined by an ideal $I$ and $I(X)$ for the ideal of polynomials vanishing at
all the points of a subset $X$ of the affine space. Again by the Closure
Theorem, we have that
$\overline{X(\mathcal{P})}=V(I(\mathcal{D}_{\mathcal{P}})\cap\mathbb{C}[{\overline{x}}])$.
Observe that $X(\mathcal{P})$ is the image of $\mathcal{D}_{\mathcal{P}}$ by
the projection
$(t,\,\overline{\Delta}\,,{\overline{x}},z)\mapsto{\overline{x}}$. This means
that $I(X(\mathcal{P}))=I(\overline{X(\mathcal{P})})\supset
I(\mathcal{D}_{\mathcal{P}})\cap\mathbb{C}[{\overline{x}}]$. Therefore,
surjectivity is equivalent to the image of $\mathcal{D}_{\mathcal{P}}$ by the
projection being Zariski closed. Given this fact, our main tool will be the
Extension Theorem:

######  Theorem 2.5.

[Extension Theorem, see [3] Ch. 3.1, Th. 3] Let $I=\langle
f_{1},\ldots,f_{s}\rangle\subseteq\mathbb{C}[x_{1},\ldots,x_{n}]$ and let
$\widetilde{I}=I\cap\mathbb{C}[x_{2},\ldots,x_{n}]$. For each $1\leq i\leq s$,
write

$f_{i}=c_{i}(x_{2},\ldots,x_{n})x_{1}^{N_{i}}+\mbox{ terms in which $x_{1}$
has degree $<N_{i}$},$

where $N_{i}\geq 0$ and $c_{i}\neq 0$. Suppose that we have a partial solution
$(a_{2},\ldots,a_{n})\in V(\widetilde{I})$. If there exists $c_{i}$ which is
nonzero at $(a_{2},\ldots,a_{n})$ then there exists $a_{1}\in\mathbb{C}$ such
that $(a_{1},a_{2},\ldots,a_{n})\in V(I)$.

Theorem 2.5 can be used to lift any partial solution ${\overline{x}}$ of
$I(\mathcal{D}_{\mathcal{P}})\cap\mathbb{C}[{\overline{x}}]$ to
$\mathcal{D}_{\mathcal{P}}$, proving surjectivity.

In the case of rational curve parametrizations, to check whether the rational
parametrization is surjective it suffices to check that for one of the
components the degree of the numerator is larger than its denominator’s. In
order to replicate this result for radical parametrizations, we will provide
$\mathbb{C}[t,\,\overline{\Delta}\,]$ with a structure of graded ring with
rational weights for the variables.

######  Definition 2.6.

First we define $\deg(t)=1$. Then we define recursively the degree of
$\Delta_{i}$ as the degree of $g_{i}$ divided by $e_{i}$, with the classical
definition of degree for a polynomial on several weighted variables; recall
that $\Delta_{i}^{e_{i}}=g_{i}$.

######  Example 2.7.

This example illustrates how the condition on the degrees that works in the
rational case is not sufficient in the radical case. Consider the
parametrization ${\mathcal{P}(t)}=(0,t-\sqrt{t^{2}-1})$ of the vertical axis
$x=0$ in $\mathbb{C}^{2}$. If we take $\Delta_{1}$ to be a square root of
$g_{1}(t)=t^{2}-1$, the degree of the polynomial corresponding to the $y$
coordinate, $y(t,\Delta_{1})=t\pm\Delta_{1}$, would be 1 in any case. Since
the denominator of $y$ is constant, the desired condition on the degrees is
true. However, the origin does not correspond to any value of $t\in\mathbb{C}$
with any of its two possible square roots.

We need to exclude some expressions in order to find easy to check sufficient
conditions of surjectivity. In order to apply the Extension Theorem to
polynomials that will arise from eliminating the $\Delta_{i}$ by multiplying
conjugates, we need conditions to ensure that we can control the degrees of
those products. Note that one can eliminate the $\Delta_{i}$ with the set
$\\{E_{1},\ldots,E_{m}\\}$, which is a triangular Gröbner basis with respect
to the lexicographical ordering $\Delta_{m}>\cdots>\Delta_{1}>t$.

######  Definition 2.8.

For every $f\in\mathbb{C}[t,\,\overline{\Delta}\,]$, we denote as $N(f)$ the
normal form of $f$ with respect to $\\{E_{1},\ldots,E_{m}\\}$. Note that the
normal form is just the polynomial obtained by repeatedly substituting every
instance of $\Delta_{i}^{e_{i}}$ by $g_{i}$.

If $f\in\mathbb{C}[t]$, observe that $N(f)=f$.

######  Example 2.9 (continuation of Example 2.7).

The problem in Example 2.7 is that $t=\infty$ corresponds to an affine point
in the parametrization. There we have two leading monomials, $t$ and
$\sqrt{t^{2}-1}$, that in a way cancel each other. In fact, to reach such
cancellation with radicals, one usually multiplies by the conjugate. For
${f(t,\Delta_{1})}=t-\Delta_{1}$ and its conjugate
$f(t,-\Delta_{1})=t+\Delta_{1}$, the normal form of their product (recall
$E_{1}=\Delta_{1}^{2}-(t^{2}-1)$) is
$N(t^{2}-\Delta_{1}^{2})=\mathrm{rem}(t^{2}-\Delta_{1}^{2},E_{1})=1$. In terms
of radicals, we can write $(t-\sqrt{t^{2}-1})(t+\sqrt{t^{2}-1})=1$. Note that
we lose degree when, after multiplying by all conjugates of $f$, we apply the
equality $E_{1}=0$ to subtitute $\Delta_{1}^{2}$ in terms of $t$.

##  3 Sufficient condition for surjectivity

Let us denote by $\mathcal{A}_{\mathcal{P}}$ the Zariski closure of the
projection of $\mathcal{D}_{\mathcal{P}}$ via the map
$(t,\,\overline{\Delta}\,,\overline{x},z)\mapsto(t,\,\overline{\Delta}\,,\overline{x})$.
We will apply repeatedly the Extension Theorem (Theorem 2.5) to lift a point
${\overline{x}}$ in $\overline{X(\mathcal{P})}$ (i.e. a solution of
$I(\mathcal{D}_{\mathcal{P}})\cap\mathbb{C}[{\overline{x}}]=I(\mathcal{A}_{\mathcal{P}})\cap\mathbb{C}[{\overline{x}}]$),
first to $\mathcal{A}_{\mathcal{P}}$, and later to
$\mathcal{D}_{\mathcal{P}}$. In the first step, we will control the
coefficients of the generator of the ideal
$I(\mathcal{A}_{\mathcal{P}})\cap\mathbb{C}[{x_{1}},t]$ (and similarly for
$x_{i}$ with $i>1$) so that for every point of $\overline{X(\mathcal{P})}$
there exists a corresponding value of $t$ (see Theorem 3.8). To this end we
will consider a product involving the conjugates of the numerator of the first
component of $\mathcal{P}$ whose degree should be preserved upon
simplification of roots. This is the motivation for the next definition. From
here onwards we will denote the imaginary unit with an upright $\mathrm{i}$ as
opposed to the italic $i$ for a variable.

######  Definition 3.1.

For $k=1,\ldots,m$, denote $\gamma_{k}=\exp(2\pi\mathrm{i}/e_{k})$. Let
$f\in\mathbb{C}[t,\,\overline{\Delta}\,]$. We define
$f_{m}=f(t,\Delta_{1},\ldots,\Delta_{m})$ and recursively for $k=m,\ldots,1$,

$F_{k}=\prod_{j=1}^{e_{k}}f_{k}(t,\Delta_{1},\ldots,\Delta_{k}\cdot\gamma_{k}^{j}),$
(3) $\quad
f_{k-1}=\mathrm{rem}(F_{k},E_{k})\in\mathbb{C}[t,\Delta_{1},\ldots,\Delta_{k-2}][\Delta_{k-1}]\mbox{
(see Lemma \ref{lema:unavariablemenos})}.$ (4)

Then we define the normalized reminder of $f$ as $R(f)=f_{0}\in\mathbb{C}[t]$.

If $f\in\mathbb{C}[t]$, observe that $R(f)=f$.

######  Lemma 3.2.

With the notation of Definition 3.1,
$f_{k-1}\in\mathbb{C}[t,\Delta_{1},\ldots,\Delta_{k-1}]$.

###### Proof.

By construction,

$F_{k}(t,\Delta_{1},\ldots,\Delta_{k})=F_{k}(t,\Delta_{1},\ldots,\gamma_{k}\Delta_{k}).$

Writing $F_{k}$ as a polynomial in $\Delta_{k}$, namely
$F_{k}=\sum_{h}a_{h}\cdot\Delta_{k}^{h}$, we have
$a_{h}=a_{h}\cdot\gamma_{k}^{h}$. Therefore, if $e_{k}$ does not divide $h$,
$\gamma_{k}^{h}\neq 1$ so $a_{h}=0$. It follows that $F_{k}$ is a polynomial
in $t,\Delta_{1},\ldots,\Delta_{k-1},\Delta_{k}^{e_{k}}$:

$F_{k}=\sum_{l}a_{le_{k}}\Delta_{l}^{le_{k}},$

so its remainder when dividing by $E_{k}$ with respect to $\Delta_{k}$ is:

$f_{k-1}=\sum_{l}a_{le_{k}}g_{k}(t,\Delta_{1},\ldots,\Delta_{k-1})^{l}.$

######  Remark 3.3.

  1. 1.

$f_{k-1}$ is the resultant of $f_{k}$ and $E_{k}$ with respect to
$\Delta_{k}$, since the resultant is the product of the evaluations of $f_{k}$
in the roots of $E_{k}$. In other words, $R(f)$ is obtained by nested
resultants.

  2. 2.

We defined $f_{k-1}$ as a remainder by a polynomial but using the normal form
instead one obtains the same $R(f)$.

  3. 3.

In general, $R(f)$ is a power of the norm of $f(t,\,\overline{\delta}\,)$ as
an algebraic element over $\mathbb{C}(t)$.

  4. 4.

If the radicals are not nested, i.e. every $g_{k}\in\mathbb{C}[t]$, we can
substitute the recursive definition by the product

$R(f)=N\left(\prod_{i_{1},\ldots,i_{m}}f(t,\gamma_{1}^{i_{1}}\Delta_{1},\ldots\gamma_{m}^{i_{m}}\Delta_{m})\right).$

This is not true in general, as the next example illustrates.

######  Example 3.4.

Consider $E_{1}=\Delta_{1}^{2}-t$, $E_{2}=\Delta_{2}^{2}-\Delta_{1}-1$ and
$f=\Delta_{1}\Delta_{2}+t$. If we compute the whole product instead of taking
remainders after conjugating each variable separately, we obtain

$N\left(\prod
f(t,\pm\Delta_{1},\pm\Delta_{2})\right)=(-2t^{3}+2t^{2})\Delta_{1}+t^{4}-t^{3}+t^{2}.$

Following Definition 3.1, we get $R(f)=t^{4}-3t^{3}+t^{2}$.

Since conjugation does not change the degree, from (3), we get
$\deg(F_{k})=e_{k}\cdot\deg(f_{k})$; on the other hand
$\deg(\Delta_{k})=\deg(g_{k})/e_{k}$ by Definition 2.6, so reducing $F_{k}$ by
$E_{k}$ (i.e. substituting $\Delta_{k}^{e_{k}}=g_{k}$ in $F_{k}$) will not
change the degree in the general case. Therefore, the degree of $R(f)$ is, at
most, $\deg(f)\cdot e_{1}\cdots e_{m}$, with equality as the expected case.
However, Example 2.9 shows that this is not always true, due to cancellations
upon reduction. Now we introduce a definition for the exceptions.

######  Definition 3.5.

Let $f\in\mathbb{C}[t,\,\overline{\Delta}\,]$. We define $f$ to be guilty when

$\deg(f)\cdot e_{1}\cdots e_{m}>{\deg(R(f))}.$

######  Remark 3.6.

By definition, no polynomial in $\mathbb{C}[t]$ is guilty.

######  Example 3.7 (continuation of Example 2.9).

We have

$f_{1}=f=t-\Delta_{1},\ F_{1}=(t-\Delta_{1})(t+\Delta_{1}),\ f_{0}=R(f)=1.$

Since $\deg(f)=1$ and $e_{1}=2$, $f$ is guilty. If we had started instead with
$y=t-\sqrt{t-1}$, we would have had the same $f=t-\Delta_{1}$ but now with
$E_{1}=\Delta_{1}^{2}-(t-1)$, so that $R(f)=t^{2}-t+1$ and $f$ would not be
guilty; one can also check that the parametrization would be surjective.

With the previous definitions we pretend to control the “image of infinity” as
in rational parametrizations. However, there is an extra issue that may appear
in with radical, which is the posibility of indeterminations of type $0/0$.
These happen for those values of $t,\,\overline{\Delta}\,$ satisfying the
equations $\,\overline{E}\,$, $p_{i}$ and $q_{i}$, where $i$ indicates the
component of $\mathcal{P}$ where the indetermination occurs. Now we can state
the main theorem.

######  Theorem 3.8.

Let $\mathcal{P}$ be as in (1), with $N(p_{i})=p_{i}$ and $N(q_{i})=q_{i}$ for
$i=1,\ldots,n$, and $\,\overline{E}\,$ be the sequence of defining polynomials
of the radical expressions. Suppose that:

  1. 1.

there exists $i$ such that $p_{i}$ is not guilty and $\deg p_{i}>\deg q_{i}$;

  2. 2.

for all $i$, the ideal
$I_{i}(\mathcal{P}):=\langle\,\overline{E}\,,p_{i},q_{i}\rangle$ is the whole
$\mathbb{C}[t,\,\overline{\Delta}\,]$.

Then the parametrization is surjective.

###### Proof.

Recall that $\mathcal{A}_{\mathcal{P}}$ is the Zariski closure of the
projection
$(t,\,\overline{\Delta}\,,\overline{x},z)\mapsto(t,\,\overline{\Delta}\,,\overline{x})$
of $\mathcal{D}_{\mathcal{P}}$. We will repeatedly apply the Extension Theorem
(Theorem 2.5) to lift a point
${\overline{x}}\in\overline{X(\mathcal{P})}=V(I(\mathcal{D}_{\mathcal{P}})\cap\mathbb{C}[{\overline{x}}])$,
firstly, to $\mathcal{A}_{\mathcal{P}}$, and, later, to
$\mathcal{D}_{\mathcal{P}}$.

Without loss of generality, assume that hypothesis 1 holds for $i=1$. Let
$F_{1}({x_{1}},t)$ be the generator of the ideal
$I(\mathcal{A}_{\mathcal{P}})\cap\mathbb{C}[{x_{1}},t]$. This makes sense
because each component of $\mathcal{A}_{\mathcal{P}}$ has dimension one (see
[8, Th. 3.11]) and its projection to the variables $({x_{1}},t)$ is finite (in
[8, Proof of Th. 3.11], the
$(t,\,\overline{\Delta}\,,\overline{x},z)\mapsto(t,\,\overline{\Delta}\,,\overline{x})$
of $\mathcal{D}_{\mathcal{P}}$ is finite and, since the dimension is one, the
missing points of the
$(t,\,\overline{\Delta}\,,\overline{x},z)\mapsto(t,\,\overline{\Delta}\,,\overline{x})$
are a finite set). Let

$G_{1}({x_{1}},t)=R({x_{1}}q_{1}-p_{1}).$

By construction $G_{1}\in I(\mathcal{A}_{\mathcal{P}})$ so $F_{1}$ divides
$G_{1}$. In Lemma 3.14 below we show that, if we consider $G_{1}$ as a
polynomial in $x_{1}$, its constant coefficient has greater degree in $t$ than
the other coefficients. Therefore the leading coefficient of $G_{1}$ with
respect to $t$ is constant, and the same thing happens to $F_{1}$. Then, every
point of $\overline{X(\mathcal{P})}$ can be lifted to a value of $t$ by
Theorem 2.5. But since each polynomial $E_{i}=\Delta_{i}^{e_{i}}-g_{i}$ is
monic with respect to $\Delta_{i}$ and they are in
$I(\mathcal{A}_{\mathcal{P}})$, every point can be lifted to the
$\,\overline{\Delta}\,$ as well.

Finally, consider a point ${\overline{x}_{0}}$ in the curve and all its lifts
to the values $t_{0},\,\overline{\Delta}\,_{0}$. We need to lift to the
variable $z$ via the condition $z\cdot\mathrm{lcm}({q_{1},\ldots,q_{n}})=1$.

Since $t_{0},\,\overline{\Delta}\,_{0},{\overline{x}_{0}}$ satisfy the
equations $q_{i}x_{i}-p_{i}=0$ for all $i\in\\{1,\ldots,n\\}$ and the
$\,\overline{E}\,$ equations, if any
$q_{i}(t_{0},\,\overline{\Delta}\,_{0})=0$ we would also have
$p_{i}(t_{0},\,\overline{\Delta}\,_{0})=0$ contradicting hypothesis 2.

######  Remark 3.9.

Let
$\mathcal{P}(t)=(p_{1}(t)/q_{1}(t),\ldots,p_{n}(t)/q_{n}(t))\in\mathbb{C}(t)^{n}$
be a rational parametrization in reduced form; that is, $\gcd(p_{i},q_{i})=1$
for all $i\in\\{1,\ldots,n\\}$. Let us assume that there exists
$i_{0}\in\\{1,\ldots,n\\}$ such that $\deg(p_{i_{0}})>\deg(q_{i_{0}})$. Then
$\mathcal{P}(t)$ satisfies the hypotheses of Theorem 3.8.

Indeed, by Def. 2.8, it holds that $N(p_{i})=p_{i}$ and $N(q_{i})=q_{i}$ for
all $i$. By Remark 3.6, $p_{i_{0}}$ is not guilty and $p_{i_{0}}/q_{i_{0}}$
satisfies hypothesis (1). Concerning hypothesis (2) one has that, since
$\gcd(p_{i},q_{i})=1$, then $1\in I_{i}(\mathcal{P})=\langle
p_{i},q_{1}\rangle$. So
$I_{i}(\mathcal{P})=\mathbb{C}[t,\,\overline{\Delta}\,]$.

######  Corollary 3.10.

Let $\mathcal{P}$ be as in (1), with all its components being reduced
polynomials in $\mathbb{C}[t,\,\overline{\Delta}\,]$ (i.e. $N(p_{i})=p_{i}$
and $q_{i}=1$ for $i=1,\ldots,n$). If hypothesis (1) in Theorem 3.8 is
satisfied, then $\mathcal{P}$ is surjective.

###### Proof.

Let us see that the second hypothesis of Theorem 3.8 is satisfies. Let
$p_{i}\in\mathbb{C}[t,\,\overline{\Delta}\,]$ be the $i$-th component of
$\mathcal{P}$, then
$I_{i}(\mathcal{P})=\langle\overline{E},p_{i},1\rangle=\langle
1\rangle=\mathbb{C}[t,\,\overline{\Delta}\,]$.

######  Corollary 3.11.

Let $\mathcal{P}$ be such that some of its components are rational and one of
its rational components satisfies the degree condition. If every non-rational
components $p_{i}/q_{i}$ is polynomial in
$\mathbb{C}[t,\,\overline{\Delta}\,]$, and $N(p_{i})=p_{i}$, then
$\mathcal{P}$ is surjective.

###### Proof.

Taking into account Def. 2.8, and the hypothesis on the non-rational
components of $\mathcal{P}$, we have that $N(p_{i})=p_{i}$ and
$N(q_{i})=q_{i}$ for all $i$. Also, reasoning as in Remark 3.9, we know that
the hypothesis (1) of Theorem 3.8 holds. Now the result follows from Corollary
3.10.

######  Remark 3.12.

The curve of $\mathbb{C}^{n}$ defined by the complex polynomials

$\\{y_{i}^{n_{i}}-g_{i}(x)\\}_{i=1,\ldots,n-1},$

and $n_{i}\in\mathbb{N}$, can be surjectively parametrized as

$\mathcal{P}=(t,\Delta_{1},\ldots,\Delta_{n-1})$

where $E_{i}:=\Delta_{i}^{n_{i}}-g_{i}(t)$.

We observe that $N(\Delta_{i})=\Delta_{i}$. Thus, the claim follows from
Corollary 3.11.

######  Remark 3.13.

As particular examples of the previous remark, we have the plane curves
$x^{n}+y^{n}=a$, with $a\in\mathbb{C}$.

######  Lemma 3.14.

With the notation and hypotheses of Theorem 3.8, let $f=xq-p$. Then, abusing
the notation of Definition 3.1, for every $k=0,\ldots,m$, the constant
coefficient of $f_{k}$ with respect to $x$ has degree $e_{k+1}\cdots
e_{m}\cdot\deg(p_{1})$ which is strictly greater than that of the other
coefficients. In particular, this applies to $G_{1}=R(f)=f_{0}$.

###### Proof.

We apply descending induction on $k$; the case $k=m$ is trivial by hypothesis.

Assume it true for $k$, that is, if $f_{k}=a_{r}x^{r}+\cdots+a_{0}$ with
$a_{j}\in\mathbb{C}[t,\Delta_{1},\ldots,\Delta_{k}]$ then
$\deg(a_{0})=e_{k+1}\cdots e_{m}\cdot\deg(p_{1})>\deg(a_{j})$ for $j>0$. Let

$F_{k}=\prod_{i=1}^{e_{k}}\left(a_{r}(t,\Delta_{1},\ldots,\Delta_{k}\cdot\gamma_{k}^{i})x^{r}+\cdots+a_{0}(t,\Delta_{1},\ldots,\Delta_{k}\cdot\gamma_{k}^{i})\right)=b_{re_{k}}x^{re_{k}}+\cdots+b_{0}$

Observe that $\deg(b_{0})=\deg(a_{0})\cdot e_{k}=e_{k}\cdots
e_{m}\cdot\deg(p_{1})$, the last equality by induction hypothesis.

For $j>0$, $b_{j}$ is a sum of products of exactly $e_{k}$ of the conjugated
$a$’s, not all being $a_{0}$. Since the degree of $a_{0}$ is strictly greater
than the degree of the others, all summands in $b_{j}$ have degree strictly
less than $e_{k}\cdot\deg(a_{0})=\deg(b_{0})$.

Now define $\tilde{b}_{j}$ such that
$\tilde{b}_{j}(t,\Delta_{1},\ldots,\Delta_{k-1},\Delta_{k}^{e_{k}})=b_{j}$.
Then

$f_{k-1}=\tilde{b}_{re_{k}}(t,\Delta_{1},\ldots,\Delta_{k-1},g_{k})x^{re_{k}}+\cdots+\tilde{b}_{0}(t,\Delta_{1},\ldots,\Delta_{k-1},g_{k}).$

The constant term of $f_{k-1}$ above coincides with the $k-1$ step in the
computation of $R(p_{1})$, possibly up to a sign. Since $p_{1}$ is not guilty,
in each reduction step for $R(p_{1})$ the degree does not drop. This shows
that

$\deg(\tilde{b}_{0}(t,\Delta_{1},\ldots,\Delta_{k-1},g_{k}))=\deg(b_{0})>\deg(b_{j})\geq\deg(\tilde{b}_{j}(t,\Delta_{1},\ldots,\Delta_{k-1},g_{k})).$

##  4 On the hypotheses for surjectivity

Deciding whether a polynomial is guilty or not can be done by defining and
checking a sort of leading coefficient for our polynomials. We will see this
for $m=1$ (i.e. $\,\overline{\Delta}\,=(\Delta_{1})$). The generalization to
$m>1$ is natural in the unnested case (i.e.
$g_{i}(t,\Delta_{1},\ldots,\Delta_{i-1})\in\mathbb{C}[t]$). For the nested
case, see Remark 4.2.

Let us denote $g(t)=g_{1}(t)=a_{0}+\cdots+a_{k}t^{k}$, $\Delta=\Delta_{1}$,
$\delta=\delta_{1}$. Consider an already reduced polynomial

$f(t,\Delta)=c_{0}(t)+c_{1}(t)\Delta+\cdots+c_{e-1}(t)\Delta^{e-1}\in\mathbb{C}[t,\Delta].$

Since we defined $\deg(\Delta)=k/e$, naming $\partial
c_{i}:=\deg_{t}(c_{i}(t))$, we have that for $i\in\\{0,\ldots,e-1\\}$

$M:=\deg(f(t,\Delta))=\max\\{\deg(c_{i}(t)\Delta^{i})\\}=\max\left\\{\partial
c_{i}+\frac{k}{e}i\right\\}.$

According to Remark 3.3,

$R(f)=R(t)=\operatorname{Resultant}_{\Delta}(f(t,\Delta),\Delta^{e}-g(t))=\prod_{r=0}^{e-1}f(t,\gamma^{r}\delta(t)),$

with $\gamma=\exp(2\pi\mathrm{i}/e)$. Now, define

$J=\\{i\in\\{0,\ldots,e-1\\}\,\,/\,\,\partial c_{i}+\frac{k}{e}i=M\\}$

and

$f_{l}(\Delta)=\sum_{i\in J}c_{i\partial
c_{i}}\Delta^{i}\in\mathbb{C}[\Delta],$

where $c_{i\partial c_{i}}$ is the leading coefficient of
$c_{i}(t)=c_{i0}+c_{i1}t+\cdots+c_{i\partial c_{i}}t^{\partial c_{i}}$. For
any factor of $R(t)$

$f(t,\gamma^{r}\delta)=c_{0}(t)+c_{1}(t)\gamma^{r}\sqrt[e]{g(t)}+\cdots+c_{e-1}(t)\gamma^{r(e-1)}\sqrt[e]{(g(t))^{e-1}},$

its coefficient of degree $M$ is

$\sum_{i\in J}\,c_{i\partial c_{i}}\gamma^{ri}(a_{k})^{\frac{i}{e}}$

(recall $a_{k}$ is the leading coefficient of $g(t)$). With this notation, we
state the following result.

######  Lemma 4.1.

With the above notation, the following are equivalent:

  1. 1.

$f(t,\Delta)$ is not guilty.

  2. 2.

For all $r\in\\{0,\ldots,e-1\\}$, $\displaystyle{\sum_{i\in J}\,c_{i\partial
c_{i}}\gamma^{ri}a_{k}^{i/e}\neq 0}$.

  3. 3.

$\operatorname{Resultant}_{\Delta}(f_{l}(\Delta),\Delta^{e}-a_{k})\neq 0$.

###### Proof.

Let $L=\displaystyle{\prod_{r=0}^{e-1}\sum_{i\in J}\,c_{i\partial
c_{i}}\gamma^{ri}a_{k}^{i/e}}=\operatorname{Resultant}_{\Delta}(f_{l}(\Delta),\Delta^{e}-a_{k}))$.
The right equality proves $(2)\Leftrightarrow(3)$.

We observe that, as seen in the proof of Lemma 3.2, $R(f)$ is the substitution
$\Delta^{e}=g(t)$ in the product of all $f(t,\gamma^{r}\Delta)$, which depends
on $t$ and $\Delta^{e}$. The highest degree homogeneous component (with our
definition of degree) of such polynomial is the product of the conjugates of
the $M-th$ homogeneous component of $f(t,\Delta)$:

$A(t,\Delta^{e})=\prod_{r=0}^{e-1}\sum_{i\in J}c_{i\partial c_{i}}t^{\partial
c_{i}}\gamma^{ri}\Delta^{i}.$

On the other side, and reasoning likewise,
$\operatorname{Resultant}_{\Delta}(f_{l}(\Delta),\Delta^{e}-a_{k}))$ is the
substitution $\Delta^{e}=a_{k}$ in the product of all
$f_{l}(\gamma^{r}\Delta)$:

$B(\Delta^{e})=\prod_{r=0}^{e-1}\sum_{i\in J}\,c_{i\partial
c_{i}}\gamma^{ri}\Delta^{i}.$

Since $B(a_{k})=L$,

$R(f)=\operatorname{Resultant}_{\Delta}(f(t,\Delta),\Delta^{e}-g(t))=Lt^{M\cdot
e}+\mbox{lower degree terms},$

which means that $(1)$ is equivalent to $(3)$.

######  Remark 4.2.

Lemma 4.1 should be easily extendable to the case with not nested radicals
and, with some more work, maybe, to the general case. However the notation
needed for the definition of $f_{l}(\,\overline{\Delta}\,)$, mainly in the
general case, would be cumbersome.

The following examples show that the hypotheses in Theorem 3.8 are sufficient
but not necessary.

######  Example 4.3.

On the condition of guiltiness, consider the radical parametrization
$x=y=t(\sqrt{t}-\sqrt{t+1})$. It can be checked that it is surjective,
although it is defined by $p_{1}=t(\Delta_{1}-\Delta_{2})$ with
$\Delta_{1}^{2}=t$, $\Delta_{2}^{2}=t+1$ which is guilty. More in detail, the
polynomial that relates $t$ and $x$ (what we called $G_{1}$ in the proof of
Theorem 3.8) is

$t^{4}-4x^{2}t^{3}-2x^{2}t^{2}+x^{4},$

which is monic in $t$ of degree 4, while from the theorem the expected degree
would be $\frac{3}{2}\cdot 4=6$; the degree drop is due to the reduction
occurring by multiplication of the four conjugates. However, since it is a
monic polynomial, we can apply Theorem 2.5 for any value of $x$.

######  Example 4.4.

On the degree condition, the Bernoulli lemniscate
$(x^{2}+y^{2})^{2}=x^{2}-y^{2}$ can be parametrized by

$x=\frac{t+t^{3}}{1+t^{4}},\quad y=\frac{t-t^{3}}{1+t^{4}}$

which is surjective although the degree condition is not fulfilled. See [6]
for more details.

######  Example 4.5.

Consider the parametrization $\left(\frac{t(\sqrt{t}-1)}{t-1},\
\frac{t^{2}+1}{t-1}\right)$. The pair $t=1,\Delta_{1}=1$ is a zero of the
ideal $I_{1}(\mathcal{P})=\langle\Delta_{1}^{2}-t,t(\Delta_{1}-1),t-1\rangle$,
so hypothesis 2 is not satisfied. However, this is not a problem with respect
to surjectivity. Indeed, since hypothesis 1 is satisfied, every point of
$\overline{X(\mathcal{P})}$ can be lifted to $t,\,\overline{\Delta}\,$; after
this, even though we do not have hypothesis 2 to guarantee the lifting to $z$
for $t=1$, there is no affine point in that situation because the second
component becomes infinity for that parameter value.

######  Remark 4.6.

The solutions of the ideal $I_{i}(\mathcal{P})$ are instances of
indeterminations of the type $0/0$ for the $i$-th component. Since the
subideal generated by $\,\overline{E}\,$ has dimension 1 in
$\mathbb{C}[t,\,\overline{\Delta}\,]$, three things can happen to the solution
set of $I_{i}(\mathcal{P})$:

  * •

it is empty, which corresponds by Hilbert’s Nullstellensatz to the part of
hypothesis 2 regarding the $i$-th component;

  * •

it is finite;

  * •

it is unidimensional. This indicates that both $p_{i}$ and $q_{i}$ become
identically zero for some component of the unidimensional
$V(\,\overline{E}\,)$. Since $\,\overline{E}\,$ is a Gröbner basis by itself
and both $p_{i}$ and $q_{i}$ are assumed to be in normal form with respect to
$\,\overline{E}\,$, this cannot happen when $V(\,\overline{E}\,)$ is
irreducible (i.e. when the subideal generated by $\,\overline{E}\,$ is
primary).

The rest of the section discusses more restrictive conditions than guiltiness
that are, on the other hand, easier to check. That is, we can substitute
hypothesis (2) of Theorem 3.8 with something else that still provides
surjectivity.

######  Proposition 4.7.

With the previous notations, if $f\in\mathbb{C}[t,\,\overline{\Delta}\,]$ and
$(t_{0},\,\overline{\Delta}\,_{0})\in\mathbb{C}^{m+1}$ satisfy
$f(t_{0},\,\overline{\Delta}\,_{0})=0$ and
$E_{i}(t_{0},\,\overline{\Delta}\,_{0})=0$ for every $i$, then
$R(f)(t_{0})=0$.

###### Proof.

By hypothesis $f_{m}=f$ vanishes at the point. Since each $F_{k}$ is a
multiple of $f_{k}$ and each $f_{k-1}$ is the remainder of $F_{k}$ by $E_{k}$,
it follows that all these polynomials vanish as well. But $R(f)=f_{0}$.

As a consequence, we have the following condition, which is computationally
more convenient, for instance with resultants.

######  Corollary 4.8.

If $\gcd(R(q_{i}),R(p_{i}))=1$ for all $i=1,\ldots,n$, then hypothesis 2 of
Theorem 3.8 holds.

Next we offer another approach. Since checking guiltiness involves considering
potentially too many conjugates and reducing, the following definition is
convenient in a computational sense.

######  Definition 4.9.

We say, again recursively, that a polynomial
$f\in\mathbb{C}[t,\,\overline{\Delta}\,]$ is suspicious when either of these
occurs:

  1. 1.

there are at least two terms of highest degree; or

  2. 2.

for some $i$, $\Delta_{i}$ appears in the (only) leading term and
$g_{i}(t,\Delta_{1},...,\Delta_{i-1})$ is suspicious.

Note that, when $g_{i}$ is suspicious, the polynomial $\Delta_{i}$ is also
suspicious.

######  Example 4.10.

Consider the expression

$\sqrt{t^{2}-1}\sqrt{t-\sqrt{t^{2}-1}}+3.$

Naming $\delta_{1}=\sqrt{t^{2}-1}$ and $\delta_{2}=\sqrt{t-\delta_{1}}$, we
have that $f(t,\Delta_{1},\Delta_{2})=\Delta_{1}\Delta_{2}+3$, which has
degree $3/2$. It is a suspicious polynomial, since
$g_{2}(t,\Delta_{1})=t-\Delta_{1}$ is clearly suspicious and $\Delta_{2}$
appears in the leading monomial.

However, $f(t,\Delta_{1},\Delta_{2})+t^{2}$ is not suspicious because its
degree is 2 and the highest degree homogeneous component only has the term
$t^{2}$.

######  Remark 4.11.

If the radicals are not nested, then the $g_{i}$ cannot be suspicious. In
particular this happens if the radical parametrization is defined with only
one root.

######  Theorem 4.12.

If a polynomial is not suspicious then it is not guilty.

###### Proof.

Let $f$ be not suspicious. With the notations of Definition 3.1, we will prove
the following by descending induction: every $f_{k}$ is not suspicious and
$\deg(f_{k-1})=e_{k}\cdot\deg(f_{k})$. The case $k=m$ is trivial. Now we
suppose it true for $k$ and prove it for $k-1$.

For any $g\in\mathbb{C}[t,\,\overline{\Delta}\,]$ we will denote as $C(g)$ its
homogeneous component of highest degree. Then
$C(f_{k})=at^{\alpha}\Delta_{1}^{\beta_{1}}\cdots\Delta_{k}^{\beta_{k}}$ where
$a\in\mathbb{C}$, $\alpha\geq 0$ and for every $i\in\\{1,\ldots,k\\}$, if
$g_{i}$ is suspicious then $\beta_{i}=0$.

We consider two cases:

  * •

If $g_{k}$ is suspicious, $\beta_{k}=0$, so $C(F_{k})=C(f_{k})^{e_{k}}$. When
passing to $f_{k-1}$, the substitution of $\Delta_{k}^{e_{k}}=g_{k}$ does not
affect $C(F_{k})$ and cannot increase the degree of lower terms because
$\deg(\Delta_{k}^{e_{k}})=\deg(g_{k})$, so $C(f_{k-1})=C(F_{k})$.

  * •

If $g_{k}$ is not suspicious,

$C(F_{k})=a^{e_{k}}t^{\alpha{e_{k}}}\Delta_{1}^{\beta_{1}e_{k}}\cdots\Delta_{k-1}^{\beta_{k-1}e_{k}}\left(\prod_{i=1}^{e_{k}}\gamma_{k}^{i}\right)^{\beta_{k}}\Delta_{k}^{\beta_{k}e_{k}}.$

Using again that the substitution of $\Delta_{k}^{e_{k}}=g_{k}$ does not
increase degrees,

$C(f_{k-1})=a^{e_{k}}t^{\alpha{e_{k}}}\Delta_{1}^{\beta_{1}e_{k}}\cdots\Delta_{k-1}^{\beta_{k-1}e_{k}}\left(\prod_{i=1}^{e_{k}}\gamma_{k}^{i}\right)^{\beta_{k}}C(g_{k})^{\beta_{k}}.$

But since $g_{k}$ is not suspicious, $C(g_{k})$ only has one monomial not
involving suspicious variables. Therefore $f_{k-1}$ is not suspicious. On the
other hand,

$\deg(f_{k-1})=\alpha{e_{k}}+\deg(\Delta_{1})\beta_{1}e_{k}+\cdots\deg(\Delta_{k-1})\beta_{k-1}e_{k}+\deg(g_{k})\beta_{k}$

and since $\deg(g_{k})=e_{k}\deg(\Delta_{k})$,

$\deg(f_{k-1})=e_{k}\left(\alpha+\deg(\Delta_{1})\beta_{1}+\cdots\deg(\Delta_{k})\beta_{k}\right)=e_{k}\deg(f_{k}).$

We conclude $\deg(f_{0})=\deg(R(f))=\deg(f)\cdot e_{1}\cdots e_{m}$.

######  Corollary 4.13.

Theorem 3.8 is also true if “not guilty” is replaced by “not suspicious”.

##  5 On missing points

Recall that we call missing points those $\overline{x}$ in the finite set
$\overline{X(\mathcal{P})}\setminus X(\mathcal{P})$ (see Definition 2.2), i.e.
points for which there do not exist
$(t,\,\overline{\Delta}\,)\in\mathbb{C}^{m+1}$ such that
$E_{i}(t,\,\overline{\Delta}\,)=0$ for all $i$ and
$\mathcal{P}(t,\,\overline{\Delta}\,)$ is well defined and equal to
$\overline{x}$.

To find missing points, let us apply projective elimination techniques as in
[3, Section 8.5]:

  1. 1.

Consider the ideal
$I\subset\mathbb{C}[t,\,\overline{\Delta}\,,{\overline{x}},z]$ generated by
the equations defining $\mathcal{D}_{\mathcal{P}}$ (see Section 2).

  2. 2.

Let $I^{h}\subset\mathbb{C}[{\overline{x}}][w,t,\,\overline{\Delta}\,,z]$ be
the ideal generated by the $(w,t,\,\overline{\Delta}\,,z)$-homogenization of
the elements of $I$ (a method based on Gröbner bases to compute $I^{h}$ can be
found in [3, Section 8.5, Proposition 10]).

  3. 3.

Note that $\overline{\mathcal{D}_{\mathcal{P}}}\,=V(I^{h})$, where
$\overline{\mathcal{D}_{\mathcal{P}}}\,$ is the projective closure of
$\mathcal{D}_{\mathcal{P}}$ in ${\mathbb{P}}^{m+2}\times{\mathbb{C}^{n}}$ (see
[3, Section 8.5, Proposition 8]). Then, by [3, Section 8.5, Corollary 9], the
projection of
$\overline{\mathcal{D}_{\mathcal{P}}}\,\subset{\mathbb{P}}^{m+2}\times{\mathbb{C}^{n}\to\mathbb{C}^{n}}$
is exactly $\overline{X(\mathcal{P})}$, so it is Zariski closed in
${\mathbb{C}^{n}}$.

  4. 4.

Therefore, the missing points are contained in the projection of the points of
$\overline{\mathcal{D}_{\mathcal{P}}}\,$ at infinity; that is, the projection
of $\overline{\mathcal{D}_{\mathcal{P}}}\,\cap
V(w)=\overline{\mathcal{D}_{\mathcal{P}}}\,\setminus\mathcal{D}_{\mathcal{P}}$
onto the $\overline{x}$ coordinates is a finite superset of the set of missing
points.

This is also a way to compute missing points for rational curves. However, it
is simpler to compute the limit of the parametrization at infinity, see [6].
We want to replicate the latter for radical parametrizations. It is worth
observing that some missing points are related to condition 1 of Theorem 3.8
(the one which allows us to lift to $\mathcal{A}_{\mathcal{P}}$, see beginning
of Section 3) and the others are related to condition 2 (the one which allows
the point in $\mathcal{A}_{\mathcal{P}}$ to be lifted to
$\mathcal{D}_{\mathcal{P}}$).

######  Remark 5.1.

The points that may not be lifted from $\overline{X(\mathcal{P})}$ to
$\mathcal{A}_{\mathcal{P}}$ are those for which the leading coefficients
$c_{\alpha_{i}}(x_{i})$ of
$F_{i}(x_{i},t)=c_{\alpha_{i}}(x_{i})t^{\alpha_{i}}+\cdots$ (the generator of
the ideal $I(\mathcal{A}_{\mathcal{P}})\cap\mathbb{C}[x_{i},t]$) vanish
simultaneously.

There are at most $\deg(c_{\alpha_{1}})\cdots\deg(c_{\alpha_{n}})$ candidates
$\overline{x}$ to be missing points. Moreover,
$V(c_{\alpha_{1}}(x_{1}),\ldots,c_{\alpha_{n}}(x_{n}))\cap\overline{X(\mathcal{P})}$
is a superset of missing points due to condition 1 of Theorem 3.8.

######  Remark 5.2.

Consider the “Castle” (union of all tower varieties of [8])
$V(\,\overline{E}\,)$ in the $m+1$-dimensional space of coordinates
$(t,\,\overline{\Delta}\,)$. The rational map
$(t,\,\overline{\Delta}\,)\mapsto\mathcal{P}(t,\,\overline{\Delta}\,)$ defined
in $V(\,\overline{E}\,)$ will be called $\mathcal{P}$, abusing the notation.
$V(\,\overline{E}\,)$ is of pure dimension one ([8]) so, since the fundamental
locus of a rational map has codimension greater than or equal to 2 in smooth
algebraic varieties, $\mathcal{P}$ can be extended to the whole
desingularization $\mathcal{C}$ of the projective closure of
$V(\,\overline{E}\,)$.

The points that are not lifted from $\overline{X(\mathcal{P})}$ to
$\mathcal{A}_{\mathcal{P}}$ are the images by $\mathcal{P}$ of the points of
$\overline{X(\mathcal{P})}$ that are “images” by $\mathcal{P}$ of points at
infinity of $V(\,\overline{E}\,)$. The degree of $V(\,\overline{E}\,)$ is, at
most, $\varepsilon_{1}\cdots\varepsilon_{m}$, where
$\varepsilon_{i}=\max\\{e_{i},\deg(g_{i})\\}$ with the classical definition of
degree, since it is an affine complete intersection of hypersurfaces of
degrees $\varepsilon_{1},\ldots,\varepsilon_{m}$. Therefore, the points at
infinity of $V(\,\overline{E}\,)$ are, counted with multiplicity,
$\varepsilon_{1}\cdots\varepsilon_{m}$. This proves that a bound for the
amount of missing points due to the lack of hypothesis 1 is
$\varepsilon_{1}\cdots\varepsilon_{m}$.

######  Example 5.3.

Consider
$\mathcal{P}=\left(\frac{\sqrt[n]{t(t-1)^{n-1}}}{t-1},\frac{\sqrt[m]{(2t-1)(t-1)^{m-1}}}{t-1}\right)=\left(\frac{\Delta_{1}}{t-1},\frac{\Delta_{2}}{t-1}\right)$,
with $E_{1}=\Delta_{1}^{n}-t(t-1)^{n-1}$,
$E_{2}=\Delta_{2}^{m}-(2t-1)(t-1)^{m-1}$. Note that $\mathcal{P}$ can be
simplified to
$\left(\sqrt[n]{\frac{t}{t-1}},\sqrt[m]{\frac{2t-1}{t-1}}\right)$, and this
imposes a different way to choose $\,\overline{\Delta}\,$ and
$\,\overline{E}\,$.

It is easy to see that $\overline{X(\mathcal{P})}$ is defined by the equation
$x^{n}-y^{m}+1=0$. The polynomials $F_{1}$ and $F_{2}$ of Remark 5.1 are

$\begin{array}[]{l}F_{1}(x,t)=x^{n}(t-1)-t=(x^{n}-1)t-x^{n},\\\
F_{2}(y,t)=y^{m}(t-1)-2t+1=(y^{m}-2)t-y^{m}+1.\end{array}$

Then $\deg(c_{\alpha_{1}})=n$ and $\deg(c_{\alpha_{2}})=m$. The possible
missing points are those whose first coordinate is an $n$-th root of $1$ and
second coordinate is an $m$-th root of $2$. One can check that, indeed, all
those points cannot be reached for any value of $t$, so we have $mn$ missing
points in total. Therefore, the bounds in Remarks 5.1 and 5.2 are sharp.

Note that $F_{1}(x,1)=-1$, hence no point in $\mathcal{A}_{\mathcal{P}}$ has
coordinate $t=1$, so there are no additional missing points.

######  Remark 5.4.

In Example 5.3, it is not difficult to check that the parametrization is
proper (i.e. generically injective). In general, when $X(\mathcal{P})$ is
irreducible, one could deduce properness if the greatest common divisor of
$F_{1}$ and $F_{2}$, seen as polynomials with coefficients in the field of
rational functions of $X(\mathcal{P})$, is the power of a degree one
polynomial.

######  Remark 5.5.

The points that cannot be lifted from $\mathcal{A}_{\mathcal{P}}$ to
$\mathcal{D}_{\mathcal{P}}$ are among those whose coordinates
$(t,\,\overline{\Delta}\,)$ satisfy the equations
$E_{1},\ldots,E_{m},p_{i},q_{i}$ for some $i=1,\ldots,n$. Taking their
projections on the space of $\,\overline{x}\,$ coordinates, this means that
the number of missing points due to hypothesis 2 failure is less than or equal
to the number of solutions of those algebraic systems of equations.

Acknowledgements. The authors are partially supported by the grant
PID2020-113192GB-I00/AEI/ 10.13039/501100011033 (Mathematical Visualization:
Foundations, Algorithms and Applications) from the Spanish State Research
Agency (Ministerio de Ciencia e Innovación). J. Caravantes and C. Villarino
belong to the Research Group ASYNACS (Ref. CT-CE2019/683). D. Sevilla is a
member of the research group GADAC and is partially supported by Junta de
Extremadura and Fondo Europeo de Desarrollo Regional (GR21055).

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# Identification of leaky Lamb waves for waveguides sandwiched between elastic
half-spaces using the Spectral Collocation Method

Evripides Georgiades<EMAIL_ADDRESS>Michael J.S. Lowe
Richard V. Craster  Also at: Department of Mathematics, Imperial College
London, London, SW7 1AY, United Kingdom  Department of Mechanical Engineering,
Imperial College London, London, SW7 1AY, United Kingdom

###### Abstract

In non-destructive evaluation guided wave inspections, the elastic structure
to be inspected is often embedded within other elastic media and the ensuing
leaky waves are complex and non-trivial to characterise; we consider the
canonical example of an elastic waveguide surrounded by other elastic
materials that demonstrates the fundamental issues with characterising the
leaky waves in such systems. Due to the complex wavenumber solutions required
to represent them, leaky waves pose significant challenges to existing
numerical methods, while methods that spatially discretise the field to
retrieve them suffer from the exponential growth of their amplitude far into
the surrounding media. We present a spectral collocation method yielding an
accurate and efficient identification of these modes, leaking into elastic
half-spaces. We discretise the elastic domains and, depending on the exterior
bulk wavespeeds, select appropriate mappings of the discretised domain to
complex paths, in which the numerical solution decays and the physics of the
problem are preserved. By iterating through all possible radiation cases, the
full set of dispersion and attenuation curves are successfully retrieved and
validated, where possible, against the commercially available software
DISPERSE. As an independent validation, dispersion curves are obtained from
finite element simulations of time-dependent waves using Fourier analysis.

##  I Introduction

Characterising leaky guided waves, and in particular accurately identifying
their attenuation and dispersion, is crucial for developing effective
practical ultrasonic inspection techniques that use guided waves for non-
destructive evaluation (NDE), e.g. in the assessment of structural integrity.
In the context of interest here, the structure under inspection comprises an
elastic waveguide in contact with dissimilar elastic media; energy is no
longer necessarily confined within the boundaries of a waveguide but rather
leaks into the adjacent media and causes the guided propagating wave to
attenuate. This behaviour is common in a wide range of settings, some of which
extends beyond structural integrity, including reinforced concrete structures
[1], seismology [2], adhesive joints and fiber composites [3, 4], buried pipes
[5, 6, 7, 8, 9] and biological structures such as bones surrounded by tissue
[10, 11, 12] Consequently, understanding the dispersion and attenuation
characteristics of leaky systems, and in particular leaky Lamb waves [13, 14,
15, 16], for waveguides in contact with spaces of different elastic materials
is of recurrent and broad interest.

Leaky Lamb waves are guided along an elastic waveguide, and are not confined
solely within the guide and continually leak energy from the elastic waveguide
into adjacent elastic media that are taken here to be of infinite extent. The
modal solutions often display the, initially puzzling, amplitude of the leaked
energy exponentially growing with distance into the adjacent elastic media
[17, 3, 14]. This is a consequence of energy conservation and this phenomenon
is discussed in detail[16] for the case of acoustic energy radiation from an
elastic waveguide immersed in fluid. For leaky wave modes, their exponential
growth in amplitude and the complex wavenumber required to describe them, both
pose a significant challenge for conventional numerical solvers used to model
guided elastic wave propagation. For instance, partial wave root finding
methods (PWRF) are well suited for solving wave propagation problems of real
wavenumber solutions, i.e. non-radiating problems, however they struggle to
converge to the attenuated waves characterised by a complex wavenumber [18].
Other methods, such as Semi Analytical Finite Element models, require
elaborate techniques to solve the nonlinear eigenvalue problems [19, 20, 21]
that emerge or require a carefully defined perfectly matched layer to
accurately model wave propagation in an infinite medium [17, 22].

An alternative to traditional methods is the use of Spectral Collocation
Methods[23, 24] (SCMs) for the discretisation of the solution domains and SCM
has had considerable success for magnetoelastic [25] or elastic guided waves
[26] in layered media, anisotropic and viscoelastic media [27, 28, 29, 30].
Recently, for a waveguide immersed with an inviscid fluid, Kiefer et. al  [14]
employed an SCM, and exact interface conditions, to account for the infinite
fluid domain in contact with the waveguide; the non-linear eigenvalue problem
obtained from this discretisation was subsequently linearised using a complex
map. In a similar manner, Tang et. al [31] created a non-linear eigenvalue
problem for a waveguide in contact with a different fluid on either side,
which they then linearised into a quintic polynomial eigenvalue problem by
moving to a higher dimensional state space: It would be a major challenge to
attempt to implement this in elasticity for a different medium in each side of
the waveguide.

Using a different approach [16], we identified the leaky Lamb waves emitted
from an infinite waveguide fully immersed in inviscid fluids also using SCMs
and overcame the challenges of an exponentially growing solution by mapping
the discretised fluid domains to complex paths, in which the numerical
solution decayed, while maintaining the physics of wave propagation in the
physical space. Here we extend to the waveguide not in contact with a fluid
but rather with elastic half-spaces. As elastic materials support both
longitudinal and shear waves, it is necessary to implement a suitable
discretisation approach to accommodate both these wave types; this is a step-
change in complexity from fluid loading cases, as either one or both
longitudinal/shear waves can exponentially grow for leaky waves. We will
demonstrate a SCM, in which the discretisation of the exterior elastic domains
is determined by the nature of the solution to be found. This method delivers
complete spectra of the dispersion curves and their mode shapes, with high
accuracy, and as “black box" solutions: there is no need for priori knowledge
of the nature of the solutions nor for the tuning of parameters to approach
the solution. Additionally, we present a quantitative validation of our
results against the commercially available PWRF software DISPERSE [32], as
well as qualitative validation of our results using Finite Element (FE)
modelling.

This paper is organised as follows - Section II presents a summary of the
underlying theory of leaky Lamb waves radiating from a waveguide in contact
with two different isotropic elastic half-spaces. Following that, Section III
presents our discretisation method and FE model, while a discussion and the
validation of our results can be found in Section IV.

##  II Theory

We consider an isotropic, linear-elastic waveguide, of finite thickness $2d$
and infinite extent in both the $x$ and $z$ directions, that is sandwiched
between two half-spaces of different isotropic and linear-elastic media. Both
the elastic waveguide and half-spaces are assumed to be homogeneous with
densities $\rho$, $\rho_{1}$ and $\rho_{2}$, longitudinal wavespeeds $c_{l}$,
$c_{l_{1}}$ and $c_{l_{2}}$ and transverse wavespeeds $c_{t}$, $c_{t_{1}}$ and
$c_{t_{2}}$ respectively. A pictorial representation of the cross section of
the setup is shown in Fig. 1.

Figure 1: Schematic of an elastic waveguide, sandwiched between two elastic
half-spaces, of thickness $2d$ and infinite extent in the $x$ and $z$
directions. The densities of the elastic media of the setup are $\rho$,
$\rho_{1}$ and $\rho_{2}$, their longitudinal wavespeeds are $c_{l}$,
$c_{l_{1}}$ and $c_{l_{2}}$ and their transverse wavespeeds are $c_{t}$,
$c_{t_{1}}$ and $c_{t_{2}}$ respectively.

The axis configuration is shown in Fig. 1 with waves assumed propagating in
the positive $x$ direction. Energy radiation into the adjacent half-spaces is
expressed mathematically in terms of a common complex wavenumber of
propagation, denoted here by $k_{x}$, whose real part reveals information
about the velocity of propagation and whose imaginary part corresponds to the
attenuation of the propagated waves. In each of the solid half-spaces, the
radiated waves are regarded as the superposition of two partial waves, one
longitudinal and one shear, whose complex wavenumbers along the $y$-direction
are $k_{y_{l_{j}}}$ and $k_{y_{t_{j}}}$ respectively, for $j=1,2$ for each of
the two elastic half-spaces. Under this regime, for a given angular frequency,
$\omega$, the following holds true [33]

$k_{y_{l_{j}}}^{2}=k_{l_{j}}^{2}-k_{x}^{2},\text{
}k_{y_{t_{j}}}^{2}=k_{t_{j}}^{2}-k_{x}^{2},\text{
}k_{y_{l_{j}}}=\frac{\omega}{c_{l_{j}}}\text{ and
}k_{y_{t_{j}}}=\frac{\omega}{c_{t_{j}}}$ (1)

These propagating and radiating waves, with displacements in only the $x$ and
$y$ directions, are the leaky Lamb waves [16] we seek.

To represent displacement fields, we denote by $\boldsymbol{\bar{u}}$ the
vector displacement in the waveguide

$\boldsymbol{\bar{u}}=\begin{pmatrix}\bar{u}_{x}\\\ \bar{u}_{y}\\\
0\end{pmatrix},$ (2)

whereby $\bar{u}_{x}$ and $\bar{u}_{y}$ are the displacements in the
propagation and transverse directions.

Similarly, for the $x-y$ plane displacement fields in the two elastic half-
spaces, we employ a Helmholtz decomposition [34, 35], expressing the vector
displacements in the two media, $\boldsymbol{\bar{u}}_{1}$,
$\boldsymbol{\bar{u}}_{2}$ as

$\boldsymbol{\bar{u}}_{j}=\begin{pmatrix}\bar{u}_{x_{j}}\\\ \bar{u}_{y_{j}}\\\
0\end{pmatrix}=\nabla\bar{\phi}_{j}+\nabla\times\begin{pmatrix}0\\\ 0\\\
\bar{\Psi}_{j}\end{pmatrix}.$ (3)

In this decomposition, $\bar{\phi}_{j}$ are the longitudinal wave potentials
in the spaces while $\bar{\Psi}_{j}$ designates the $z-$component of their
transverse wave potentials.

Here, each of $\bar{u}_{x},\bar{u}_{y},\bar{\phi}_{j}$ and $\bar{\Psi}_{j}$
are functions of $x,y$ and $t$, while, when plane harmonic waves are assumed
to be propagating through the structure, the common exponential term
$\exp(i(k_{x}x-\omega t))$ fully describes their harmonic nature. Thus, they
decompose into

$\bar{\phi}_{j}(x,y,t)={\phi}_{j}(y)\exp\left[i(k_{x}x-\omega t)\right],$ (4)

with ${\phi}_{j}(y)$ representing the $y-$dependent components of the
longitudinal wave potentials in the half-spaces and similarly for
$\bar{u}_{x}(x,y,t),\bar{u}_{y}(x,y,t)$ and $\bar{\Psi}_{j}(x,y,t)$.

With this notation, the governing equations of motion, resulting from momentum
equations [16, 33], are given by

$\displaystyle\rho\omega^{2}u_{x}-(\lambda+2\mu)k_{x}^{2}u_{x}+(\lambda+\mu)ik_{x}\frac{\partial
u_{y}}{\partial y}+\mu\frac{\partial^{2}u_{x}}{\partial y^{2}}$
$\displaystyle=0,$ (5) $\displaystyle\rho\omega^{2}u_{y}-\mu
k_{x}^{2}u_{y}+(\lambda+\mu)ik_{x}\frac{\partial u_{x}}{\partial
y}+(\lambda+2\mu)\frac{\partial^{2}u_{y}}{\partial y^{2}}$ $\displaystyle=0,$
(6)
$\displaystyle\rho_{j}\omega^{2}\phi_{j}-k_{x}^{2}(\lambda_{j}+2\mu_{j})\phi_{j}+(\lambda_{j}+2\mu_{j})\frac{\partial^{2}\phi_{j}}{\partial
y^{2}}$ $\displaystyle=0,$ (7)
$\displaystyle\rho_{j}\omega^{2}\Psi_{j}-k_{x}^{2}\mu_{j}\Psi_{j}+\mu_{j}\frac{\partial^{2}\Psi_{j}}{\partial
y^{2}}$ $\displaystyle=0,$ (8)

with $\lambda$ and $\mu$, and $\lambda_{j}$ and $\mu_{j}$, the elastic Lamé
parameters of the waveguide and the two elastic half-spaces respectively.

The physics of the problem require the continuity of tractions and
displacements at the two interfaces. That is to say, that at the two
interfaces $y_{1}=-d$ and $y_{2}=d$,
$\sigma_{y}-\sigma_{y_{j}}=0,\sigma_{xy}-\sigma_{xy_{j}}=0,u_{x}-u_{x_{j}}=0$
and $u_{y}-u_{y_{j}}=0$, with $\sigma_{y_{j}}$ and $\sigma_{y_{j}}$ and
$\sigma_{xy}$ and $\sigma_{xy_{j}}$ the normal and shear stresses in the
waveguide and half-spaces respectively [33, 34].

These are explicitly given in Eqs. (9)-(11)

$\displaystyle\begin{split}&\left.\left[(\lambda+2\mu)\frac{\partial
u_{y}}{\partial y}+\lambda
ik_{x}u_{x}+\lambda_{j}k_{x}^{2}\phi_{j}\right]\right|_{y=y_{j}}+\\\
&\left.\left[-(\lambda_{j}+2\mu_{j})\frac{\partial^{2}\phi_{j}}{\partial
y^{2}}-2\mu_{j}ik_{x}\frac{\partial\Psi_{j}}{\partial
y}\right]\right|_{y=y_{j}}=0,\end{split}$ (9)
$\displaystyle\begin{split}&\left.\left[\mu\left(ik_{x}u_{y}+\frac{\partial
u_{x}}{\partial y}\right)\right]\right|_{y=y_{j}}+\\\
&\left.\left[-\mu_{j}\left(2ik_{x}\frac{\partial\phi_{j}}{\partial
y}-k_{x}^{2}\Psi_{j}-\frac{\partial^{2}\Psi_{j}}{\partial
y^{2}}\right)\right]\right|_{y=y_{j}}=0,\end{split}$ (10)
$\displaystyle\left.\left[u_{x}-ik_{x}\phi_{j}+\frac{\partial\Psi_{j}}{\partial
y}\right]\right|_{y=y_{j}}=0,$ (11)
$\displaystyle\left.\left[u_{y}-\frac{\partial\phi_{j}}{\partial
y}-ik_{x}\Psi_{j}\right]\right|_{y=y_{j}}=0.$ (12)

##  III Method

###  III.1 Discretisation

A spectral collocation discretisation [36, 23], based on Chebyshev
polynomials, is applied to obtain a discrete representation of both the
waveguide and the open spaces of the elastic solids.

Wave solutions of the setup described in the previous section are generally
split into two types - non-radiating and leaky modes. In our setup, the
waveguide is in contact with solids; hence the media surrounding the waveguide
support shearing as well as compressive motion. Consequently, in addition to
the non-radiating, evanescent, wave solutions, we can further split the leaky
modes solutions in two cases - those that have purely shear (shear leaky) or
shear and compressive (fully leaky) polarisations. For these cases, the
exponential growth of the amplitude of leaky waves is attributed to the
exponential growth of some or all of the partial waves depicted in Fig. 1. In
particular, when shear leaky waves are to be considered in a solid half-space,
the shear partial wave and corresponding transverse wave potential,
$\Psi_{j}$, exponentially grow in amplitude, while the longitudinal partial
wave and corresponding wave potential, $\phi_{j}$, are evanescent. In the
fully leaky case, both partial waves and their wave potentials in the solid,
exhibit exponential growth.

Let us consider the structure of Fig. 1, with $c_{l_{1}}\geq
c_{l_{2}}>c_{t_{1}}\geq c_{t_{2}}$. In the region where the phase velocity of
the propagating wave, $c_{ph}$ is less than $c_{t_{2}}$, evanescent waves in
both solids are expected, while when it is between $c_{t_{1}}$ and
$c_{t_{2}}$, shear leaky waves in the bottom elastic solid and evanescent at
the top are expected. When $c_{ph}$ is between $c_{l_{2}}$ and $c_{t_{1}}$,
shear leaky waves in both spaces are expected, while when it is between
$c_{l_{1}}$ and $c_{l_{2}}$, the shear leaky waves of the bottom solid are now
expected to be fully leaky. Finally, when $c_{ph}$ is greater than
$c_{l_{1}}$, fully leaky waves are expected to be radiating in both solid
half-spaces. An analogous analysis can be deduced for other combinations of
longitudinal and shear bulk wavespeeds. An illustration of this is shown in
Fig. 2. It should finally be noted that some exceptions of this rule, namely
of subsonic radiation, have been recorded in the literature [37, 38, 39, 14].
Such examples are still obtainable by our method, however we believe that
further study is required with regards to their physical properties, and they
are therefore not considered here.

Figure 2: Illustration of the different radiation cases with an increase in
phase velocity for the exemplar case of $c_{l_{1}}\geq c_{l_{2}}>c_{t_{1}}\geq
c_{t_{2}}$.

For the simpler case with external fluids, and a single wavespeed in the
exterior, the numerical issues associated with exponential growth of the
solution are discussed and a method is introduced to counter the numerical
instabilities inherent to the growth [16]. As shown in Fig. 2, with elastic
half-spaces adjacent to a waveguide and all media having both shear and
longitudinal wavespeeds, an exponential growth of the solution is prominent in
most but not all cases. As a result, a discretisation catering to this growth
is necessary. Here we choose to solve for each of the cases described above
separately, considering whether each of the partial waves of the solution
exponentially grows in one or both elastic spaces. When this is the case we
judiciously implement an analogous discretisation to the one described in [16]
for each of Eqs. (7) and (8), provided that the corresponding wave potential
exponentially grows. We comment that this separation into several solutions
does not indicate a necessity to know the nature of the guided modes a priori;
all we are doing here is composing the full solution from the several
contributions of solutions for these cases. The nature of the unknown guided
wave modes then determines when they emerge as solutions as we go through the
calculations of the different cases.

We start by utilising the Differentiation Matrix (DM) suite of Weideman and
Reddy [40] to produce first and second order DMs, denoted by $D^{(1)}$ and
$D^{(2)}$ respectively, based on Chebyshev polynomials for the $N$ Gauss-
Lobatto Chebyshev collocation points

$s_{i}=-\cos\left(\frac{(i-1)\pi}{N-1}\right),$ (13)

for $i=1,\dots,N$ and $s_{i}\in[-1,1]$.

These provide an approximation of first and second order derivatives of
functions defined over $[-1,1]$, evaluated at the discrete set
$\\{s_{i}\\}_{i=1}^{N}$. We need the DMs and the discretisation of the domains
of definition of our displacements and wave potentials, which are $[-d,d]$ for
the displacements $u_{x}$ and $u_{y}$, and the half-spaces $[-\infty,-d]$ and
$[d,\infty]$ for the potentials $\phi_{1}$ and $\Psi_{1}$ and $\phi_{2}$ and
$\Psi_{2}$ respectively. To achieve this, mappings that map the collocation
points $\\{s_{i}\\}_{i=1}^{N}$ of $[-1,1]$ to collocation points in the
domains given above are used. The corresponding DMs are then the result of the
application of the chain rule.

A linear map, that is, a simple multiplication by $d$, maps the collocation
points from $[-1,1]$ to $[-d,d]$, providing a discretisation of the closed
domain of the waveguide. As for the discretisation of the domains of
definition of the wave potentials in the two elastic solid half-spaces, the
rational maps

$\displaystyle h_{\phi_{1}}:s\longrightarrow
y=-d-\zeta_{\phi_{1}}\frac{1-s}{1+s},$ (14) $\displaystyle
h_{\psi_{1}}:s\longrightarrow y=-d-\zeta_{\psi_{1}}\frac{1-s}{1+s},$ (15)

with $\zeta_{\phi_{1}},\zeta_{\psi_{1}}\in\mathbb{R}$, map the collocation
points from $[-1,1]$ to points in $[-\infty,-d]$, while the rational maps

$\displaystyle h_{\phi_{2}}:s\longrightarrow
y=d+\zeta_{\phi_{2}}\frac{1+s}{1-s},$ (16) $\displaystyle
h_{\psi_{2}}:s\longrightarrow y=d+\zeta_{\psi_{2}}\frac{1+s}{1-s},$ (17)

with $\zeta_{\phi_{2}},\zeta_{\psi_{2}}\in\mathbb{R}$, map the collocation
points from $[-1,1]$ to points in $[d,\infty]$.

These maps and resulting DMs, provide an accurate and numerically stable
discretisation of both elastic half-spaces, provided that the solution we are
after is that of all four partial waves being evanescent in the two spaces.
However, as was discussed earlier, special care and a modification of the maps
of Eqs. (14) - (17), needs to be carried out when solving for the cases of
Fig. 2, where some or all of the partial waves and corresponding wave
potentials exponentially grow.

For clarity, we demonstrate how we modify the maps and discretise the domains
to solve for waves that radiate as shear leaky waves in one of the half-
spaces, say the bottom one, and evanescent in the other.

When a wave potential is exponentially growing, representing leaky waves in an
elastic solid half-space, a complex rational map, mapping collocation points
from $[-1,1]$ to a path in the complex plane is used. In the exemplary case of
shear leaky waves in the bottom half-space and evanescent waves in the other,
only the transverse wave potential in the bottom solid, $\Psi_{2}$,
exponentially grows. Thus, we modify only the map presented in Eq. (17) by
allowing $\zeta_{\psi_{2}}$ to be complex valued, resulting in the
discretisation of a complex path, similarly to [16], with one of the
collocation points located on the interface at $y=d$. From the chain rule, the
DMs, for functions defined over that path, are given by

$\displaystyle D_{\psi_{2}}^{(1)}=$
$\displaystyle\text{diag}\left(\frac{2\boldsymbol{\zeta}_{\psi_{2}}}{[\boldsymbol{\zeta}_{\psi_{2}}+(\boldsymbol{y}_{\psi_{2}}-\boldsymbol{d})]^{2}}\right)D^{(1)},$
(18)
$\displaystyle\begin{split}D_{\psi_{2}}^{(2)}=&\text{diag}\left(\frac{-4\boldsymbol{\zeta}_{\psi_{2}}}{[\boldsymbol{\zeta}_{\psi_{2}}+(\boldsymbol{y}_{\psi_{2}}-\boldsymbol{d})]^{3}}\right)D^{(1)}\\\
&+\text{diag}\left(\frac{4\boldsymbol{\zeta}_{\psi_{2}}^{2}}{[\boldsymbol{\zeta}_{\psi_{2}}+(\boldsymbol{y}_{\psi_{2}}-\boldsymbol{d})]^{4}}\right)D^{(2)}.\end{split}$
(19)

where
$\boldsymbol{\zeta}_{\psi_{2}}=(\zeta_{\psi_{2}},\dots,\zeta_{\psi_{2}})^{T},\boldsymbol{d}=(d,\dots,d)^{T}$
and $\boldsymbol{y}_{\psi_{2}}$ is the vector of collocation points in the
complex path as a result of the mapping given in Eq. (17).

The outgoing partial wave corresponding to the exponentially growing
transverse wave potential $\psi_{2}$ is characterised by the exponential term
$\exp[ik_{y_{t_{2}}}(y-d)]$. For $\psi_{2}$ to grow exponentially as $y$ tends
to $\infty$, $-\Im(k_{y_{t_{2}}})$ must be a positive real number. By
performing the discretisation presented here, we are mapping $y$ to a complex
path such that the exponent can be expressed as
$ik_{y_{t_{2}}}(y^{\prime}-d)$, with $y^{\prime}=\zeta_{\psi_{2}}(y-d)+d$. By
an appropriate choice of $\zeta_{\psi_{2}}$ [16], the exponential term is made
to decay.

This newly constructed, numerically decaying, function is the analytical
continuation of the transverse potential $\psi_{2}$ over the complex path
defined by Eq. (17), and we denote it by $\tilde{\psi}_{2}$. Just like
$\psi_{2}$ it is fully determined by its value at the interface, while due to
our choice of mapping, which maintains one collocation point at the physical
interface of the waveguide and the bottom solid, $\psi_{2}$ and
$\tilde{\psi}_{2}$ attain the same value at that interface. What is more, as a
time harmonic function, $\tilde{\psi}_{2}$ also satisfies the governing Eq.
(8) for the same wavenumber $k_{x}$. As a result, by discretising a complex
path as opposed to the physical half-space of the elastic solid, we are able
to solve Eq. (8) for a numerically stable and decaying function for the same
complex wavenumber $k_{x}$, while also being able to reconstruct the physical,
exponentially growing, solution from its value at the interface.

Using this method, Eq. (8) can be discretised and presented in matrix form as
follows

$\left(\rho_{j}\omega^{2}I-k_{x}^{2}\mu_{j}I+\mu_{j}D_{\psi_{2}}^{(2)}\right)\boldsymbol{\tilde{\psi}_{2}}=\boldsymbol{0},$
(20)

where $I$ is the $N\times N$ identity matrix and
$\boldsymbol{\tilde{\psi}_{2}}$ is the vector containing the evaluations of
$\tilde{\psi}_{2}$ at the collocation points in the complex path.

For the non-evanescent partial waves, we use the rational maps in Eqs.
(14)-(15) to treat Eqs. (5)-(8) in an analogous manner, i.e. to obtain a
discretisation of the domains and a matrix formulation of the equations.

Finally, similarly to [16, 26, 14], we merge all discretised equations into a
single, matrix equation, replacing the rows corresponding to the collocation
points at the two interfaces with the relevant, discretised, interface
conditions. This results in a polynomial eigenvalue problem

$(k_{x}^{2}L_{2}+k_{x}L_{1}+L_{0})\boldsymbol{u}=0,$ (21)

where
$\boldsymbol{u}=(\boldsymbol{\phi_{1}},\boldsymbol{\psi_{1}},\boldsymbol{u_{x}},\boldsymbol{u_{y}},\boldsymbol{\phi_{2}},\boldsymbol{\tilde{\psi}_{1}})^{T}$,
with each entry in the vector $\boldsymbol{u}$ being the vector of the
respective quantity evaluated at the collocation points, and where
$L_{2},L_{1}$ and $L_{0}$ are the $k_{x}^{2}$ dependent $k_{x}$ dependent and
$k_{x}$ independent components of the merged matrix equation.

We treat other cases of Fig. 2 in the same manner, each time discretising a
path in the complex plane to accommodate for the exponential growth of leaky
waves. We solve each of the resulting eigenvalue problems using companion
matrix linearisations [14, 41] and conventional eigenvalue problem solvers.

###  III.2 Finite Element Modelling

As an additional verification tool, we have used FE modelling to model wave
propagation in the waveguide of finite thickness adjacent to two elastic half-
spaces shown schematically in Fig. 1.

We used the GPU-based FE solver Pogo[42]. The equation of motion for
elasticity theory, in matrix form and with spatial discretisation using FE, is

$\mathbf{M}\mathbf{\ddot{U}}+\mathbf{C}\mathbf{\dot{U}}+\mathbf{K}\mathbf{U}=\mathbf{F},$
(22)

where $\mathbf{M}$, $\mathbf{C}$, $\mathbf{K}$ and $\mathbf{F}$ are the mass,
damping, stiffness and applied force matrices, while $\mathbf{U}$,
$\mathbf{\dot{U}}$ and $\mathbf{\ddot{U}}$ are the displacement, velocity and
acceleration matrices respectively. Pogo uses the finite-difference method
[43] in time to discretise Eq. (22) as follows:

$\textbf{M}\frac{\textbf{U}^{m+1}-2\textbf{U}^{m}+\textbf{U}^{m-1}}{\Delta
t^{2}}+\textbf{C}\frac{\mathbf{U}^{m+1}-\mathbf{U}^{m-1}}{2\Delta
t}+\textbf{KU}^{m}=\textbf{F},$ (23)

where $m$ stands for the $m^{\text{th}}$ time step and $\Delta t$ is the
duration of a single time step. Therefore, knowing the solution at two
consecutive time steps allows for the retrieval of the displacement at the
next time step by solving for $\textbf{U}^{m+1}$ in Eq. (23). The FE modelling
approach described here follows the FE models used in other elastic wave
progation studies [44, 45, 46].

Figure 3: (Colour online) A schematic of a FE domain used to model elastic
wave propagation and retrieve the dispersion characteristics of guided modes
in a waveguide adjacent to two elastic half-spaces. In the figure, source
nodes are shown in red and monitor nodes in black. Absorbing layers around the
domain are shown in orange while different colours denote the different
materials within the domain.

For the FE modelling of leaky wave propagation in a waveguide coupled to two
elastic half-spaces, a rectangular domain was used - a schematic of the cross
sectional area of the domain is shown in Fig. 3. Each element of the domain is
assigned with the material properties of the space it is a part of (Young’s
modulus, density, Poisson’s ratio), while linear, square elements of four
nodes each were used. The mesh refinement was set to approximately
$dx=\lambda/30$, to ensure for the sufficient convergence of the solution
[47]. Here, $\lambda$ is the smallest bulk wavelength amongst the three
materials, calculated at the frequency of the simulation. Stiffness-reducing
absorbing layers [48] were defined around the domain to accommodate for its
unbounded nature, with thickness equal to four times the longest bulk
wavelength.

A source line, comprised of multiple source nodes, was defined in the middle
of the domain - source nodes are nodes with predefined displacement/force, and
are used to excite the propagating wave in the model. Two Hann-windowed
tonebursts were applied to each source node, orthogonally to each other.
Depending on the nature of the desired excitation, multi-modal or single-mode
excitation, the amplitude and relative phase of the tonebursts at each source
node were determined. For a multi-modal excitation, all source nodes were
excited in phase with the same broadband (single-cycled) toneburst in both
directions. For single-mode excitation, the method discussed in [15] was
followed; first, the displacement fields (mode shapes) of the desired mode
were extracted from our method and verified with those from DISPERSE. Then,
the amplitude and phase of the mode shapes were used to modulate the
amplitudes and phases of the excitation signals accordingly.

A monitor line, comprised of multiple monitor nodes, was placed horizontally,
to the right of the source line, along the length of the domain, as shown in
Fig. 3 \- a monitor node is a node whose displacement profile is recorded and
stored for post-processing. The placement of the monitor line is dependent on
the nature of the desired mode to be observed. Leaky modes attenuate as they
propagate along the waveguide, with most of their energy observable in the
exterior solids, requiring a monitor line that is placed close to the source
line and in the exterior elastic solids. Contrary to that, non-radiating modes
remain trapped in the waveguide and can propagate longer distances; thus a
monitor line in the middle of the waveguide is more suitable for their
observation. In post-processing, a two-dimensional fast Fourier transform
(2D-FFT)[49], in space and time, is applied to the collection of the stored
monitor signals of interest. The results of the 2D-FFT were then normalised to
the maximum amplitude to obtain the dispersion characteristics of the excited
modes in a normalised amplitude scale.

A typical model size is of the order of $1\times 10^{7}$ degrees of freedom.
The models were solved using an Nvidia (Santa Clara, California) 2080 Ti
graphics card with 8GB of memory. The solution time was approximately forty
seconds. More details about the excitation, the size of the domain and the
material properties will be given in the following section as they are
specific to the example presented there.

##  IV Results and Discussion

We now utilise our SCM method to retrieve the dispersion curves and
attenuation characteristics of leaky and non-radiating modes in an elastic
waveguide in contact with two elastic solid half-spaces. The obtained
dispersion curves and attenuation plots are then verified against the
commercially-available software DISPERSE while the dispersion curves are also
validated against FE simulations of elastic wave propagation using Pogo.

We illustrate the method on the exemplary, yet industrially relevant case of
adhesive joints of two aluminium alloys, studied in [50] to demonstrate ways
of determining the material properties and thickness of the adhesive layer and
assess the quality of the joint. An epoxy adhesive is modelled as a waveguide
of finite thickness, $d=0.5$ mm, density $\rho_{\text{ep}}=1.17\text{
g/cm}^{3}$, longitudinal wavespeed $c_{l_{\text{ep}}}=2.61\text{ m/ms}$ and
transverse wavespeed $c_{t_{\text{ep}}}=1.1\text{ m/ms}$. The spaces of the
aluminium components are modelled as linear elastic half-spaces of isotropic
media, both with density $\rho_{\text{al}}=2.82\text{ g/cm}^{3}$, longitudinal
wavespeed $c_{l_{\text{al}}}=6.33\text{ m/ms}$ and transverse wavespeed
$c_{t_{\text{al}}}=3.12\text{ m/ms}$.

For the retrieval of the solution, the phase velocity domain was split up in
three sections in accordance with Fig. 2 and the wavespeeds of the adjacent
aluminium half-spaces. For each region an angular frequency sweep of 150
frequency steps, up to $\omega=30\text{ MHz}$, was performed. An
appropriate[16] choice of complex parameters
$\zeta_{\phi_{j}},\zeta_{\psi_{j}}$ was made in each sweep. This choice
effectively distributed the collocation points across real and complex paths,
ensuring a well-distributed spread of points in the domain. For the retrieval
of non-radiating solutions,
$\left(\zeta_{\phi_{j}},\zeta_{\psi_{j}}\right)=\left(10,10\right)$ was
chosen, while
$\left(\zeta_{\phi_{j}},\zeta_{\psi_{j}}\right)=\left(10,10i\right)$ were
chosen for shear leaky and
$\left(\zeta_{\phi_{j}},\zeta_{\psi_{j}}\right)=\left(10i,10i\right)$ for
fully leaky ones. In each domain, $50$ collocation points were used which,
together with the choice of the complex parameters
$\zeta_{\phi_{j}},\zeta_{\psi_{j}}$ and the complex mappings Eq. (14)-(17),
allowed for an appropriate spread of the collocation points in the exterior
domains throughout the frequency range of interest.

Subsequently, the numerical solutions obtained underwent a comprehensive
filtering process. Therein their propagating nature ($\Re(k_{x})>0$ and
$|k_{x}|<\infty$), attenuation characteristics (less than $15$ Np/mm) and
compliance with the physical interface conditions (up to 3 decimal places)
were assessed. Under this processes only modes of physical interest remain, as
back-propagating, non-propagating, highly attenuative, bulk-wave and non-
converged solutions were eliminated. The computation took a total of
approximately 18 minutes on a personal computer with 32GB RAM memory and an
i7-10700 CPU.

As a validation, the results of our SCM method were compared against those
obtained from a multi-modal excitation of our adhesive joint domain in a FE
simulation using Pogo. All source nodes in the FE model were excited in phase
with the same broadband toneburst in both the $x$ and $y$ directions. The
excitation toneburst used at each source node was the sum of sixteen single-
cycled Hann-windowed toneburtsts, each with a central frequency varying from
$0.25-4$ MHz at $0.25$ MHz increments. In this case, wavelengths and mesh
refinement, $dx$, were calculated at $4$ MHz. The domain, excluding absorbing
layers, was 11 mm thick, 1 mm of epoxy in the middle of the domain and 5 mm of
aluminium each side of the waveguide, and 100 mm in length. The monitor line
used for post-processing was placed at $y=2$ mm. Results of the described FE
model are shown in Fig. 4; the results were normalised against the maximum
amplitude of all the modes recorded.

Figure 4: (Colour online) Dispersion curves for of an epoxy waveguide in
contact with an aluminium half-space on each side, obtained by our SCM
(denoted by $\circ$), by DISPERSE (denoted by –) and by FE modelling
(normalised intensity in colour).

As shown in Fig. 4, there is good agreement between all three methods, for the
modes that the chosen forcing in the FE model was able to excite. There are
instances where DISPERSE was not able to fully retrieve a mode, or missed it
altogether, highlighting the merits of using the proposed SCM instead of root
finding techniques.

Additionally, a comparison of the attenuation values obtained by the SCM
against those of DISPERSE is shown in Fig. 5a. For visualisation purposes,
solutions with high attenuation, $\Im(k_{x})>10\text{ Np/mm}$, are not
displayed here. A detail of the region of up to 1 Np/mm is shown in 5b.

Figure 5: (Colour online) Comparison of attenuation values obtained by our SCM
(denoted by $\circ$) and by DISPERSE (denoted by –) for waves in an epoxy
waveguide in contact with an aluminium half-space on each side 5a. A detail of
the region with attenuation of up to 1 Np/mm 5b.

As Fig. 5 shows, the attenuation characteristics obtained by our SCM match
those obtained by DISPERSE, demonstrating in this way the ability of our SCM
to accurately capture the complex wavenumber solutions of the leaky modes of
the system.

Following these initial validations of the SCM results, we proceed with
presenting two cases of single-mode excitation - one where both DISPERSE and
SCM were able to retrieve the mode to be excited, and one where only SCM was
able to obtain the relevant mode. The former provides confidence in the
results of the FE model, which is subsequently used as a validation metric for
our SCM in the latter case where DISPERSE results are not available. In each
case, the excitation tonebursts were set to the same central frequency, and
for the purposes of a narrowband excitation, the number of cycles in the
excitation signal was increased to eleven. The size of the absorbing layers,
and $dx$ were calculated accordingly at the central frequency.

Fig. 6a shows the results from the excitation of a shear leaky mode, that is a
mode that only radiates energy in both exterior domains purely in the form of
shear waves, at $f=1.03$ MHz, which both DISPERSE and SCM were able to
retrieve, as well as the relevant FE result from the post-processing of
monitor signals from a monitor line placed horizontally at $y=5$ mm. The mode
shape from our SCM, used for this excitation, is also shown in Fig. 6b,
together with that of DISPERSE.

Figure 6: (Colour online) Dispersion curves showing the normalised intensity
and dispersion of the shear leaky mode excited at $f=1.03$ MHz with a single-
mode excitation, validated against SCM and DISPERSE dispersion curves 6a. The
mode shapes from the SCM that were used for the excitation are validated
against those obtained by DISPERSE 6b.

Fig. 6b demonstrates good agreement between the mode shapes of the two methods
(SCM and DISPERSE). Fig. 6a also shows that using the SCM data of Fig. 6b we
were able to independently excite the same mode using FE modelling.

Fig. 7a shows the results from the excitation of a non-radiating mode, at
$f=3.53$ MHz, which only our SCM was able to retrieve, alongside the relevant
FE result. As the energy of such modes are confined within the boundaries of
the waveguide, the monitor line of the FE model is now placed horizontally in
the middle of the waveguide. The mode shape used for this excitation, as
obtained from our SCM, is shown in Fig. 7b.

Figure 7: (Colour online) Dispersion curves showing the normalised intensity
and dispersion of the non-radiating mode, found only by SCM, and excited at
$f=3.53$ MHz with a single-mode excitation 7a. The mode shapes from the SCM
that were used for the excitation are shown in Fig. 7b.

It is worth noting that even if the model was provided with an incorrect
displacement profile, it would have converged to a valid solution[43]. Thus,
since all three methods agree on a true solution in Fig. 6a, the FE model is
able to verify the existence of a mode independently of SCM and DISPERSE.
Therefore we believe that the agreement between FE and SCM in Fig. 7a is
sufficient validation of the SCM in places where a DISPERSE solution is not
available.

##  V Conclusion

Leaky Lamb wave propagation in a waveguide that is in contact with elastic
half-spaces has been considered; these wave modes are often characterised by a
complex wavenumber, a subsequent exponential growth in amplitude far into the
surrounding elastic media and a highly attenuative nature along the waveguide,
making their retrieval by traditional numerical solvers challenging. Here, we
overcome these obstacles by formulating a theoretical framework for the
problem, discretising it using a SCM, that also accommodates for the
exponentially growing nature of the problem, and finally solving the resulting
polynomial eigenvalue problem for the retrieval of the wave mode solutions of
interest.

The results of our SCM have been verified against a commercially available
software, DISPERSE, and FE modelling. In cases where DISPERSE was able to
retrieve a solution to the problem, it was used simultaneously as a validation
tool for the SCM and the FE model. Subsequently, having verified the FE model,
it was used to produce results that validated our SCM for a case where
DISPERSE was unable to do so. The SCM approach is accurate, versatile, it
readily identifies all leaky modes in elasticity without requiring knowledge
of the modes or any tuning a priori and is easy to code; the key point is to
remove the exponential growth from the numerical scheme, whilst it remains
within the physics, and doing so via the mapping to a complex path reduces the
leaky problem to a standard form for numerical implementation.

###### Acknowledgements.

EG and RVC are supported by the EU H2020 FET Open “Boheme” Grant No. 863179

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