# Order of torsion for reduction of linearly independent points for a family
of Drinfeld modules

Dragos Ghioca  and  Igor E. Shparlinski  Dragos Ghioca
Department of Mathematics
University of British Columbia
Vancouver, BC V6T 1Z2
Canada<EMAIL_ADDRESS>Igor E. Shparlinski
School of Mathematics and Statistics
University of New South Wales
Sydney NSW 2052
Australia<EMAIL_ADDRESS>
###### Abstract.

Let $q$ be a power of the prime number $p$, let $K={\mathbb{F}}_{q}(t)$, and
let $r\geqslant 2$ be an integer. For points ${\mathbf{a}},{\mathbf{b}}\in K$
which are ${\mathbb{F}}_{q}$-linearly independent, we show that there exist
positive constants $N_{0}$ and $c_{0}$ such that for each integer
$\ell\geqslant N_{0}$ and for each generator $\tau$ of
${\mathbb{F}}_{q^{\ell}}/{\mathbb{F}}_{q}$, we have that for all except
$N_{0}$ values $\lambda\in\overline{\mathbb{F}_{q}}$, the corresponding
specializations ${\mathbf{a}}(\tau)$ and ${\mathbf{b}}(\tau)$ cannot have
orders of degrees less than $c_{0}\log\log\ell$ as torsion points for the
Drinfeld module $\Phi^{(\tau,\lambda)}:{\mathbb{F}}_{q}[T]\longrightarrow{\rm
End}_{\overline{\mathbb{F}_{q}}}({\mathbb{G}}_{a})$ (where ${\mathbb{G}}_{a}$
is the additive group scheme), given by $\Phi^{(\tau,\lambda)}_{T}(x)=\tau
x+\lambda x^{q}+x^{q^{r}}$.

###### Key words and phrases:

Drinfeld modules, reduction modulo primes, unlikely intersections

###### 2010 Mathematics Subject Classification:

Primary 37P05, Secondary 37P35

##  1\. Introduction

###  1.1. Notation

Let $q$ a power of a prime number $p$. Consequently, we denote by
${\mathbb{F}}_{q}$ the finite field with $q$ elements and also, let
$\overline{\mathbb{F}_{q}}$ be its algebraic closure. We let
$K={\mathbb{F}}_{q}(t)$ be the rational function field of transcendence degre
$1$ over ${\mathbb{F}}_{q}$. Then each ${\mathbf{a}}\in K$ can be written as a
quotient ${P(t)}/{Q(t)}$ for relatively prime polynomials
$P,Q\in{\mathbb{F}}_{q}[t]$ (with $Q\neq 0$); so, for all but finitely many
$\tau\in\overline{\mathbb{F}_{q}}$ (as long as $Q(\tau)\neq 0$), the
specialization map $t\mapsto\tau$ yields a specialization
${\mathbf{a}}\mapsto{\mathbf{a}}(\tau)\in\overline{\mathbb{F}_{q}}$.
Furthermore, for any polynomial $f\in K[x]$, then for all but finitely many
$\tau\in\overline{\mathbb{F}_{q}}$ the specialization $t\mapsto\tau$ is well-
defined for each coefficient of $f(x)$; in this case, we denote by
$f(\tau;x)\in\overline{\mathbb{F}_{q}}[x]$ the corresponding specialization of
$f(x)$.

An (${\mathbb{F}}_{q}[T]$-)Drinfeld module $\Phi$ defined over any field $L$
containing ${\mathbb{F}}_{q}$ is given by a non-linear action of
${\mathbb{F}}_{q}[T]$ on the additive group scheme ${\mathbb{G}}_{a}$, that
is, it is an endomorphism $\Phi:{\mathbb{F}}_{q}[T]\longrightarrow{\rm
End}_{L}({\mathbb{G}}_{a})$ where not all of the maps
$\Phi_{P(T)}:=\Phi(P(T))$ are linear. In particular, each such Drinfeld module
is uniquely defined by an additive polynomial (of degree larger than $1$) with
coefficients in $L$ corresponding to $\Phi_{T}$. We recall thar a polynomial
$f(x)$ is called _additive_ if $f(x+y)=f(x)+f(y)$ for all $x$ and $y$. These
are linear polynomials in characteristic zero and polynomials with monomial
terms of degrees which are powers of $p$ in positive characteristic $p$. A
point ${\mathbf{c}}\in\overline{L}$ is a torsion point for $\Phi$ if there
exists a nonzero polynomial $P(T)\in{\mathbb{F}}_{q}[T]$ such that
$\Phi_{P(T)}({\mathbf{c}})=0$; the _monic_ polynomial $P(T)$ of minimal degree
for which $\Phi_{P(T)}({\mathbf{c}})=0$ is called the _order_ of
${\mathbf{c}}$. For more details on Drinfeld modules, we refer the reader to
[Gos96].

###  1.2. Unlikely intersections

Inspired by the classical unlikely intersection problems in arithmetic
geometry (for example, see [Hab09] and the references therein), Masser and
Zannier [MZ10, MZ12] have proved a remarkable result: there exist finitely
many $\lambda\in{\mathbb{C}}$ such that both the points
$\left(2,\sqrt{2(2-\lambda)}\right)$ and $\left(3,\sqrt{6(3-\lambda)}\right)$
are torsion on the corresponding elliptic curve $E_{\lambda}$ from the
Legendre family given by the equation $y^{2}=x(x-1)(x-\lambda)$. The results
from [MZ10, MZ12] have been the starting point of a very intense research
campaign in arithmetic dynamics, which emulates the classical unlikely
intersection problems from arithmetic geometry, but this time, the unlikely
intersection taking place in an algebraic dynamics setting (for example, see
[BD11, GHT13, GKN17]). In this spirit, the first author and Hsia have proved
in [GH13, Theorem 1.5] the following result.

######  Theorem A.

Let $r>1$ be an integer, let $q$ be a power of the prime number $p$, and let
$K={\mathbb{F}}_{q}(t)$. We consider the family of Drinfeld modules
$\Phi^{(z)}$ (parametrized by $z\in\overline{K}$) such that

$\Phi_{T}^{(z)}(x)=tx+zx^{q}+x^{q^{r}}.$

Let ${\mathbf{a}},{\mathbf{b}}\in K$. Then there exist infinitely many
$z\in\overline{K}$ such that both ${\mathbf{a}}$ and ${\mathbf{b}}$ are
torsion points for the Drinfeld module $\Phi^{(z)}$ if and only if
${\mathbf{a}}$ and ${\mathbf{b}}$ are ${\mathbb{F}}_{q}$-linearly dependent.

######  Remark 1.1.

In [GH13, Theorem 1.5], the above conclusion in Theorem A has been stated
under the assumption that there exist infinitely many $\lambda$ in the
separable closure $K^{{\rm sep}}$ of $K$. On the other hand, we observe that
the main technical ingredient used in the proof of [GH13, Theorem 1.5] comes
from the equidistribution theorem of Baker and Rumely [BR10, Theorem 7.52],
which is stated for product formula fields. So, in [GH13, Theorem 2.6] (which
provides the key step in proving [GH13, Theorem 1.5]), one could work instead
with the perfect closure $K^{{\rm
per}}:={\mathbb{F}}_{q}\left(t^{1/p^{n}}\colon n\geqslant 1\right)$, which is
a product formula field since each valuation on $K$ has a unique extension to
$K^{{\rm per}}$ and thus, the hypotheses (1)-(4) from [BR10, Definition 7.51,
p. 185] are met for $K^{{\rm per}}$. Then since $\left(K^{{\rm
per}}\right)^{{\rm sep}}=\overline{K}$, one obtains our Theorem A with an
identical proof as in [GH13, Theorem 1.5].

######  Remark 1.2.

We also note that if ${\mathbf{b}}=\zeta\cdot{\mathbf{a}}$ for some
$\zeta\in{\mathbb{F}}_{q}$, then
$\Phi^{(z)}_{P(T)}({\mathbf{b}})=\zeta\cdot\Phi^{(z)}_{P(T)}({\mathbf{a}})$
for _each_ $z\in\overline{K}$; therefore, we have that ${\mathbf{a}}$ is
torsion for $\Phi^{(z)}$ if and only if ${\mathbf{b}}$ is torsion for
$\Phi^{(z)}$, and moreover, they both have the same order as torsion points
for $\Phi^{(z)}$.

One of the unlikely intersection results inspired by [MZ10, MZ12] is the
result of Habegger [Hab13] that there exist finitely many pairs
$(a,b)\in{\mathbb{C}}^{2}$ with the property that _each_ of the three points

(1.1)
$\left(1,\sqrt{1+a+b}\right),\quad\left(2,\sqrt{8+2a+b}\right),\quad\left(3,\sqrt{27+3a+b}\right)$

are torsion points for the elliptic curve $y^{2}=x^{3}+ax+b$. This motivated
the second author to prove the following result (see [Shp20, Theorem 1.1]).

######  Theorem B.

There is an absolute constant $M_{0}$ such that for all primes $p$ and for all
but at most $M_{0}$ pairs $(a,b)\in\overline{{\mathbb{F}}_{p}}^{2}$ with
$4a^{3}+27b^{2}\neq 0$ at least one of the points from (1.1) is of order at
least $(\log p)^{1/33+o(1)}$.

Theorem B is proven in [Shp20] by employing both the result of Habegger
[Hab13], but also using in an essential way [D’AOSS19, Theorem 2.1] which
provides a way to control the number of solutions to polynomial equations when
they are reduced modulo primes.

###  1.3. Our results

Here we prove the following result which is a counterpart for Drinfeld modules
of Theorem B (also, note that as observed in [Gos96, Chapter 3], the family of
Drinfeld modules $\Phi^{(z)}$ which plays the counterpart role of elliptic
curves is the one for which $\Phi^{(z)}_{T}(x)=tx+zx^{q}+x^{q^{2}}$, as we
vary $z$ in $\overline{K}$).

######  Theorem 1.3.

Let $r\geqslant 2$ be an integer, let $q$ be a power of the prime $p$, and let
$K={\mathbb{F}}_{q}(t)$. We let ${\mathbf{a}},{\mathbf{b}}\in K$ be two
${\mathbb{F}}_{q}$-linearly independent points. Then there exist positive
constants $N_{0}$ and $c_{0}$ (depending only on $q$, $r$ and the degrees of
${\mathbf{a}}$ and ${\mathbf{b}}$) such that for each $\ell\geqslant N_{0}$
and for each generator $\tau$ of the extension
${\mathbb{F}}_{q^{\ell}}/{\mathbb{F}}_{q}$, we have that for all except
$N_{0}$ values of $\lambda\in\overline{\mathbb{F}_{q}}$, at least one of the
two points ${\mathbf{a}}(\tau)$ or ${\mathbf{b}}(\tau)$ is a torsion point
whose order has degree at least $c_{0}\log\log(\ell)$ for the Drinfeld module
$\Phi^{(\tau,\lambda)}\in{\rm
End}_{\overline{\mathbb{F}_{q}}}({\mathbb{G}}_{a})$ given by
$\Phi^{(\tau,\lambda)}_{T}(x):=\tau x+\lambda x^{q}+x^{q^{r}}$.

######  Remark 1.4.

In Theorem 1.3, the family of Drinfeld modules $\Phi^{(\tau,\lambda)}$ given
by $\Phi^{(\tau,\lambda)}_{T}(x):=\tau x+\lambda x^{q}+x^{q^{r}}$ varies with
respect to both parameters $\tau$ and $\lambda$ in
$\overline{\mathbb{F}_{q}}$. Note that for any given
$\tau\in\overline{\mathbb{F}_{q}}$, one could easily find
$\lambda\in\overline{\mathbb{F}_{q}}$ such that ${\mathbf{a}}(\tau)$ is killed
by $\Phi_{T}$, say, and therefore ${\mathbf{a}}(\tau)$ could be a torsion
point of order of a very small degree for the Drinfeld module
$\Phi^{(\tau,\lambda)}$; but then the conclusion of our Theorem 1.3 yields
that ${\mathbf{b}}(\tau)$ cannot be a torsion point whose order has also very
small degee compared to $[{\mathbb{F}}_{q}(\tau):{\mathbb{F}}_{q}]$. Again,
just like it has been observed in Remark 1.2, the fact that ${\mathbf{a}}$ and
${\mathbf{b}}$ are ${\mathbb{F}}_{q}$-linearly independent is paramount since
otherwise, if ${\mathbf{b}}/{\mathbf{a}}\in{\mathbb{F}}_{q}^{*}$ (for example)
then for any suitable specialization $t\mapsto\tau$, we would have that also
${\mathbf{a}}(\tau)$ and ${\mathbf{b}}(\tau)$ are ${\mathbb{F}}_{q}$-linearly
dependent and therefore, they are torsion points for any Drinfeld module
$\Phi^{(\tau,\lambda)}$ of the same order (which thus could be very small
relative to $[{\mathbb{F}}_{q}(\tau):{\mathbb{F}}_{q}]$).

######  Remark 1.5.

The fact that in the conclusion of our Theorem 1.3 we ask that $\ell\geqslant
N_{0}$ is merely for guaranteeing that the specialization
${\mathbf{a}}\mapsto{\mathbf{a}}(\tau)$ (resp.
${\mathbf{b}}\mapsto{\mathbf{b}}(\tau)$) is well-defined for each generator
$\tau$ for the extension ${\mathbb{F}}_{q^{\ell}}/{\mathbb{F}}_{q}$ (note that
there are only finitely many $\tau\in\overline{\mathbb{F}_{q}}$ for which the
above specializations do not make sense).

######  Remark 1.6.

The extra logarithm appearing in $c_{0}\log\log\ell$ from the conclusion of
our Theorem 1.3 comes form the fact that we insist that for _each_
specialization $t\mapsto\tau$ where $\tau$ _generates_ the extension
${\mathbb{F}}_{q^{\ell}}/{\mathbb{F}}_{q}$ our desired conclusion regarding
the orders of the torsion points ${\mathbf{a}}(\tau)$ and ${\mathbf{b}}(\tau)$
must hold. Our proof could be easily modified to show that there exists _some_
$\tau$ generating the extension ${\mathbb{F}}_{q^{\ell}}/{\mathbb{F}}_{q}$
with the property that the specialization $t\mapsto\tau$ yields that at least
one of ${\mathbf{a}}(\tau)$ or ${\mathbf{b}}(\tau)$ would be a torsion point
for $\Phi^{(\tau,\lambda)}$ (for each $\lambda\in\overline{\mathbb{F}_{q}}$
apart from some $N_{0}$ values $\lambda$) whose order has degree at least
$c_{1}\log(\ell)$ (for some positive constant $c_{1}$ depending only on $q$,
$r$ and the degrees of ${\mathbf{a}}$ and ${\mathbf{b}}$).

We also note that our Theorem 1.3 (similar to Theorem B of [Shp20]) fits into
a more general program of obtaining results for specialization over finite
fields for certain global results stemming from arithmetic dynamics (see also
[AG09, BCMOS20, Cha13, CKSZ14, D’AOSS19, KMS21, Mel21, Shp18]). Also, strictly
in the context of Drinfeld modules, there has been interest in the past for
considering Drinfeld modules defined over finite fields, either for studying
some of their intrinsic arithmetic properties (as in [Ghi08]), or for infering
aritmetic properties of Drinfeld modules of generic characteristic after
considering their reductions at various places (see [GT08, Hsi08]).
Furthermore, our results provide additional evidence that Drinfeld modules are
the right vehicle in positive characteristic for formulating questions which
are analogue to classical arithmetic problems in the context of abelian
varieties (see [Bre12, Ghi05, Sca02] for results in the context of Drinfeld
modules, which are analogues of the classical André-Oort, Mordell-Lang,
respectvely Manin-Mumford conjectures).

We also note that [AG09, Theorem 5.2] provides a somewhat related result to
our Theorem 1.3; however, there are some significant differences between that
result and ours. The result from [AG09, Theorem 5.2] refers to reductions at
various places of a _given_ Drinfeld module, while in our result we vary both
the specialization $t\mapsto\tau$ but also vary $\lambda$ in
$\overline{\mathbb{F}_{q}}$; in other words,
$\left\\{\Phi^{(\tau,\lambda)}\right\\}$ is a two-dimensional family of
Drinfeld modules defined over $\overline{\mathbb{F}_{q}}$. Finally, we refer
to our Section 4 for a discussion regarding further research in the direction
set by our paper.

We sketch briefly the plan for our paper. The approach to our Theorem 1.3
follows the general direction set in [Shp20] for the case of elliptic curves.
So, we need to establish a technical result (which is our Lemma 2.4) regarding
the preservation of the number of zeros in specializations of polynomials over
a function field. Then using also a couple of technical results (see Lemmas
2.7 and 2.5) regarding the growth of heights and of degrees of the iterates of
a given point under polynomials in a family of Drinfeld modules, we obtain the
desired technical ingredients for finishing the proof of our Theorem 1.3,
which is completed Section 3. We conclude with a discussion in Section 4 of
further research in the direction opened up by our article.

##  2\. Preliminary results

###  2.1. Greatest common divisors of polynomials

We recall that a greatest common divisor of polynomials
$f_{1},\ldots,f_{s}\in{\mathbb{L}}[x]$ over a field ${\mathbb{L}}$ is any
polynomial $d\in{\mathbb{L}}[x]$ such that any other common divisor $e$ of
$f_{1},\ldots,f_{s}$ divides $d$ (clearly $d$ is defined up to a nonzero
scalar multiple). In particular, we can define the degree
$\deg\gcd(f_{1},\ldots,f_{s})$, which does not depend on a particular choice
of a greatest common divisor.

We need the following special case of [vzGKS97, Lemma 4.8].

######  Lemma 2.1.

Let ${\mathbb{L}}$ be a field and let $f_{1},\ldots,f_{s}\in{\mathbb{L}}[x]$
be $s\geqslant 2$ polynomials of degree at most $D$. For any finite set
${\mathcal{A}}\subseteq{\mathbb{L}}$, uniformly chosen random elements
$\alpha_{3},\ldots,\alpha_{s}\in{\mathcal{A}}$ and

$f_{0}=f_{2}+\sum_{3\leqslant i\leqslant s}\alpha_{i}f_{i}\in{\mathbb{L}}[x].$

we have

$\operatorname{Prob}_{\alpha_{3},\ldots,\alpha_{s}\in{\mathcal{A}}}\left[\deg\gcd(f_{1},\ldots,f_{s})=\deg\gcd(f_{0},f_{1})\right]\geq
1-D/\\#{\mathcal{A}}.$

We can now estimate the degree of the coefficients in the Bézout identity for
relatively prime polynomials.

######  Lemma 2.2.

Let ${\mathbb{L}}$ be a field and let $f_{1},\ldots,f_{s}\in{\mathbb{L}}[x]$
be $s\geqslant 2$ relatively prime polynomials of degree at most $D$. There
are polynomials $b_{1},\ldots,b_{s}\in{\mathbb{L}}[x]$ of degree at most $D$
such that

(2.1) $b_{1}f_{1}+\ldots+b_{s}f_{s}=1.$

###### Proof.

Taking the algebraic closure ${\overline{\mathbb{L}}}$ of ${\mathbb{L}}$
(which is not necessary if ${\mathbb{L}}$ is infinite, or even finite but
satisfying $\\#{\mathbb{L}}>D$), we see from Lemma 2.1 that there are some
$\alpha_{3},\ldots,\alpha_{s}\in{\overline{\mathbb{L}}}$ such that $f_{1}$ and
$f_{0}$, defined as in Lemma 2.1 are relatively prime. Note that $\deg
f_{0}\leqslant D$ and hence using the standard Euclidean algorithm we see that
there are $b_{0},b_{1}\in{\overline{\mathbb{L}}}[x]$ of degrees at most $D-1$
such that

$b_{0}f_{0}+b_{1}f_{1}=1$

and we obtain (2.1) with
$b_{1},b_{2}=b_{0},b_{3}=\alpha_{3}b_{0},\ldots,b_{s}=\alpha_{s}b_{0}\in{\overline{\mathbb{L}}}[x]$.

We now observe that (2.1) defines a system of linear equations for $Ds$
coefficients of polynomials $b_{1},\ldots,b_{s}$. The above argument shows
that that system is compatible over ${\overline{\mathbb{L}}}$, and since the
coefficients of this system are defined over ${\mathbb{L}}$, then this means
that the system is compatible over ${\mathbb{L}}$ as well. Hence we can find
polynomials $b_{1},\ldots,b_{s}\in{\mathbb{L}}[x]$ satisfying (2.1). ∎

###  2.2. Preserving number of common zeros in specializations of polynomials
over a function field

######  Definition 2.3.

For a polynomial $f\in{\mathbb{F}}_{q}[t][x]$, we define its height
${\mathfrak{h}}(f)$ as the maximum degree of the coefficients of $f$, which
are themselves polynomials in ${\mathbb{F}}_{q}[t]$. (Note that for any
element in ${\mathbb{F}}_{q}(t)$, its Weil height is simply its degree as a
rational function.)

One of our main tools is the following result. Note that in our proof of
Theorem 1.3, we only employ Lemma 2.4 for two polynomials (that is, $s=2$ with
the notation as in Lemma 2.4); however, we include this more general version
of Lemma 2.4 which may be useful beyond our current application. For further
generalisations see Section 4.

######  Lemma 2.4.

Let $f_{1},\ldots,f_{s}\in{\mathbb{F}}_{q}[t][x]$ be polynomials of degree at
most $D$ and of heights at most $H$. Assume that there exist at most $N_{0}$
common zeros in $\overline{{\mathbb{F}}_{q}(t)}$ of the polynomials
$f_{1}(x),\ldots,f_{s}(x)$. Then there exists a nonzero polynomial
$\mathcal{T}\in{\mathbb{F}}_{q}[T]$ of degree at most $(2D+1)H$ with the
property that for each $\tau\in\overline{\mathbb{F}_{q}}$ such that
$\mathcal{T}(\tau)\neq 0$, for the specialization polynomials $f_{i}(x)\mapsto
f_{i}(\tau;x)$, $i=1,\ldots,s$, corresponding to the specialization
$t\mapsto\tau$, we have that the polynomials

$f_{1}(\tau;x),\ldots,f_{s}(\tau;x)\in\overline{\mathbb{F}_{q}}[x]$

have at most $N_{0}$ common zeros in $\overline{\mathbb{F}_{q}}$.

###### Proof.

Let $h\in{\mathbb{F}}_{q}[t][x]$ be the greatest primitive common divisor of
$f_{1},\ldots,f_{s}$, that is, the coefficients of $h$ are relatively prime as
polynomials in $t$. We first claim that considering $f_{1},\ldots,f_{s},h$ as
bivariate polynomials in ${\mathbb{F}}_{q}[t,x]$ we still have the
divisibility $h\mid f_{i}$, $i=1,\ldots,s$. Indeed, we know that $h$ divides
$f_{i}$, $i=1,\ldots,s$, in the ring ${\mathbb{F}}_{q}(t)[x]$ and so there are
$g_{i}\in{\mathbb{F}}_{q}[t][x]$ and $b\in{\mathbb{F}}_{q}[t]$ such that $b$
is relatively prime to $g_{1},\ldots,g_{s}$ as polynomials in
${\mathbb{F}}_{q}[t,x]$ and

$f_{i}=(g_{i}/b)\cdot h$

that is

$bf_{i}=hg_{i}$

Since $h$ is primitive and ${\mathbb{F}}_{q}[t,x]$ is a unique factorisation
domain, we deduce that $b\in{\mathbb{F}}_{q}$.

Hence for $g_{i}:=f_{i}/h$ we have $g_{i}\in{\mathbb{F}}_{q}[t][x]$,
$i=1,\ldots,s$. Furthermore, we see that $g_{1},\ldots,g_{s}$ have no common
zeros in $\overline{{\mathbb{F}}_{q}(t)}$ and hence by Lemma 2.2 there are for
some $b_{1},\ldots,b_{s}\in{\mathbb{F}}_{q}(t)[x]$ of degree at most $D-1$
such that

(2.2) $b_{1}g_{1}+\ldots+b_{s}g_{s}=1.$

We now see that (2.2) can be rewritten as a system of at most $2D$ linear
equations over ${\mathbb{F}}_{q}(t)$ in $Ds$ variables. Choosing that largest
nonsingular submatrix of the matrix defining this system, from the Cramer rule
we see that (2.2) has a solution in $b_{i}=c_{i}/\Delta$ where
$c_{i}\in{\mathbb{F}}_{q}[t][x]$, $i=1,\ldots,s$ and
$\Delta\in{\mathbb{F}}_{q}[t]$ is a polynomial of degree (in $t$)

$\deg(\Delta)\leqslant 2DH.$

We now write (2.2) as $c_{1}g_{1}+\ldots+c_{s}g_{s}=\Delta$ or

(2.3) $c_{1}f_{1}+\ldots+c_{s}f_{s}=\Delta h.$

Therefore, for any $\tau\in\overline{\mathbb{F}_{q}}$ which is not a root of
$\Delta(t)$ we derive from (2.3) that

$c_{1}(\tau;x)f_{1}(\tau;x)+\ldots+c_{s}(\tau;x)f_{s}(\tau;x)=\Delta(\tau)h(\tau;x).$

By our assumption, $h(x)$ has at most $N_{0}$ roots in
$\overline{{\mathbb{F}}_{q}(t)}$; hence, unless $h(\tau;x)$ is identically
equal to zero, then it must have at most $N_{0}$ roots in
$\overline{\mathbb{F}_{q}}$.

Since as we have observed the bivariate polynomial $h\in{\mathbb{F}}_{q}[t,x]$
divides, for example, $f_{1}\in{\mathbb{F}}_{q}[t,x]$, then we see that
${\mathfrak{h}}(h)\leqslant H$. In particular, the greatest monic common
divisor $h_{{\rm gcd}}(t)\in{\mathbb{F}}_{q}[t]$ of all the coefficients of
$h(x)$ (which are themselves polynomials in ${\mathbb{F}}_{q}[t]$) must have
degree at most $H$. Hence, letting $\mathcal{T}(t):=\Delta(t)\cdot h_{{\rm
gcd}}(t)$, we obtain a nonzero polynomial in ${\mathbb{F}}_{q}[t]$ of degree
at most $2DH+H=(2D+1)H$ which satisfies the desired conclusion from Lemma 2.4.
∎

###  2.3. Technical background on Drinfeld modules

In this Section we derive some basic results regarding the growth of the
heights of the polynomials in a family of Drinfeld modules; for more results
regarding heights for Drinfeld modules, we refer the reader to [Den92, Ghi07,
GH13, GT08].

Let $r\geqslant 2$ be an integer, and let $K:={\mathbb{F}}_{q}(t)$. We
consider the family of Drinfeld modules
$\Phi^{(z)}:{\mathbb{F}}_{q}[T]\longrightarrow{\rm
End}_{K(z)}({\mathbb{G}}_{a})$ given by
$\Phi^{(z)}_{T}(x):=tx+zx^{q}+x^{q^{r}}$ parametrized by $z\in\overline{K}$.

Let ${\mathbf{a}}\in K$ be a nonzero point; we write
${\mathbf{a}}={{\mathbf{a}}_{1}}/{{\mathbf{a}}_{2}}$ with (nonzero) relatively
prime polynomials ${\mathbf{a}}_{1},{\mathbf{a}}_{2}\in{\mathbb{F}}_{q}[t]$.
Let $D:=\deg{\mathbf{a}}$ (or equivalently, $D$ is the height of
${\mathbf{a}}$, as defined in [Ghi07]), that is,
$D:=\max\\{\deg_{t}{\mathbf{a}}_{1},\deg_{t}{\mathbf{a}}_{2}\\}$. Then for
each positive integer $n$, we have that $\Phi_{T^{n}}^{(z)}({\mathbf{a}})$ is
a polynomial in $K[z]$; for more details, see [GH13, Section 4].

More precisely, let

$f_{n}(z):=\Phi_{T^{n}}^{(z)}({\mathbf{a}}),$

then as proven in [GH13, Lemma 4.2], we have that
$\deg_{z}(f_{n}(z))=q^{r(n-1)}$ (since $f_{1}(z)={\mathbf{a}}^{q}\cdot
z+t{\mathbf{a}}+{\mathbf{a}}^{q^{r}}$ is a linear polynomial in $z$). Also,
the leading coefficient of $f_{n}(z)$ is ${\mathbf{a}}^{q^{1+r(n-1)}}$.
Furthermore, for each $P(T)\in{\mathbb{F}}_{q}[T]$, we may consider the
polynomial in $z$ given by $z\mapsto\Phi^{(z)}_{P(T)}({\mathbf{a}})$, which is
a polynomial with coefficients in $K$. In particular, we have the following
immediate corollary of [GH13, Lemma 4.2].

######  Lemma 2.5.

Let $0\neq{\mathbf{a}}\in K$ and let $P(T)\in{\mathbb{F}}_{q}[T]$ be a non-
constant polynomial. Then the map $z\mapsto\Phi^{(z)}_{P(T)}({\mathbf{a}})$ is
a polynomial in $K[z]$ of degree $q^{r(\deg(T)-1)}$.

In our proof of Theorem 1.3 we also need the following result, which gives us
control on the heights of the coefficients of $f_{n}(z)$.

######  Lemma 2.6.

Let ${\mathbf{a}}:={\mathbf{a}}_{1}/{\mathbf{a}}_{2}\in
K={\mathbb{F}}_{q}(t)$, where ${\mathbf{a}}_{1}$ and ${\mathbf{a}}_{2}$ are
nonzero coprime polynomials in ${\mathbb{F}}_{q}[t]$. As before, we let
$f_{n}(z):=\Phi_{T^{n}}^{(z)}({\mathbf{a}})\in K[z]$ for each positive integer
$n$. Then $g_{n}(z):={\mathbf{a}}_{2}^{q^{rn}}\cdot
f_{n}(z)\in{\mathbb{F}}_{q}[t][z]$ and moreover,

${\mathfrak{h}}(g_{n})\leqslant(2+\deg({\mathbf{a}}))^{q^{rn}}.$

###### Proof.

We have that $f_{1}(z)={\mathbf{a}}^{q}z+t{\mathbf{a}}+{\mathbf{a}}^{q^{r}}$
and so,

$g_{1}(z)={\mathbf{a}}_{2}^{q^{r}}\cdot
f_{1}(z)={\mathbf{a}}_{1}^{q}{\mathbf{a}}_{2}^{q^{r}-q}z+t{\mathbf{a}}_{1}{\mathbf{a}}_{2}^{q^{r}-1}+{\mathbf{a}}_{1}^{q^{r}}\in{\mathbb{F}}_{q}[t][z];$

furthermore, ${\mathfrak{h}}(g_{1})\leqslant
1+\deg({\mathbf{a}})^{q^{r}}<(2+\deg({\mathbf{a}}))^{q^{r}}$. Also, we have
that for each positive integer $n$,

$f_{n+1}(z)=f_{n}(z)^{q^{r}}+zf_{n}(z)^{q}+tf_{n}(z)$

and so,

(2.4) $\begin{split}g_{n+1}(z)&={\mathbf{a}}_{2}^{q^{r(n+1)}}f_{n+1}(z)\\\
&=g_{n}(z)^{q^{r}}+{\mathbf{a}}_{2}^{q^{r(n+1)}-q^{1+rn}}zg_{n}(z)^{q}+t{\mathbf{a}}_{2}^{q^{r(n+1)}-q^{rn}}g_{n}(z).\end{split}$

Therefore, an easy induction yields that $g_{n}(z)\in{\mathbb{F}}_{q}[t][z]$
and furthermore,
${\mathfrak{h}}(g_{n}(z))\leqslant(2+\deg({\mathbf{a}}))^{rn}$ for all
$n\in\mathbb{N}$; note that since
$\deg({\mathbf{a}}_{2})\leqslant\deg({\mathbf{a}})$, then the contribution to
the height of $g_{n+1}(z)$ of the last term in (2.4) is at most equal to

$1+\deg({\mathbf{a}})^{q^{r(n+1)}-q^{rn}}\cdot(2+\deg({\mathbf{a}}))^{q^{rn}}\leqslant(2+\deg({\mathbf{a}}))^{q^{r(n+1)}}.$

This concludes our proof of Lemma 2.6. ∎

Finally, the next result is an immediate consequence of Lemma 2.6.

######  Lemma 2.7.

Let ${\mathbf{a}}:={\mathbf{a}}_{1}/{\mathbf{a}}_{2}$ for some nonzero coprime
polynomials ${\mathbf{a}}_{1},{\mathbf{a}}_{2}\in{\mathbb{F}}_{q}[t]$. Then
for each non-constant polynomial $P(T)\in{\mathbb{F}}_{q}[T]$, the map

$z\mapsto{\mathbf{a}}_{2}^{q^{r\deg_{T}(P(T))}}\cdot\Phi^{(z)}_{P(T)}({\mathbf{a}})$

is a polynomial in ${\mathbb{F}}_{q}[t][z]$ of height at most
$(2+\deg({\mathbf{a}}))^{q^{r\deg_{T}(P(T))}}$.

##  3\. Proof of Theorem 1.3

###  3.1. Bounds on degree and height of some polynomials

So, we have ${\mathbf{a}},{\mathbf{b}}\in K={\mathbb{F}}_{q}(t)$ which are
${\mathbb{F}}_{q}$-linearly independent, and we write

${\mathbf{a}}:=\frac{{\mathbf{a}}_{1}}{{\mathbf{a}}_{2}}\qquad\mbox{and}\qquad{\mathbf{b}}:=\frac{{\mathbf{b}}_{1}}{{\mathbf{b}}_{2}}$

with nonzero relatively prime polynomials
${\mathbf{a}}_{1},{\mathbf{a}}_{2}\in{\mathbb{F}}_{q}[t]$ (resp.
${\mathbf{b}}_{1},{\mathbf{b}}_{2}\in{\mathbb{F}}_{q}[t]$). Then
$D:=\max\\{\deg({\mathbf{a}}),\deg({\mathbf{b}})\\}$ is a positive integer
since otherwise, we would have that both ${\mathbf{a}}$ and ${\mathbf{b}}$ are
elements of ${\mathbb{F}}_{q}$ and thus, they would be
${\mathbb{F}}_{q}$-linearly dependent, contradicting our hypothesis.

According to Theorem A, there exist at most $N_{0}$ points $z\in\overline{K}$
such that both ${\mathbf{a}}$ and ${\mathbf{b}}$ are torsion points for the
Drinfeld module $\Phi^{(z)}:{\mathbb{F}}_{q}[T]\longrightarrow{\rm
End}_{K(z)}({\mathbb{G}}_{a})$ given by
$\Phi^{(z)}_{T}(x)=tx+zx^{q}+x^{q^{r}}$. At the expense of replacing $N_{0}$
by a larger positive integer, we may also assume that neither
${\mathbf{a}}_{2}$ nor ${\mathbf{b}}_{2}$ has a root $\tau$ generating an
extension of ${\mathbb{F}}_{q}$ of degree at least equal to $N_{0}$.

Now, let $\ell\geqslant N_{0}$ and let $\tau$ be a generator for the finite
fields extension ${\mathbb{F}}_{q^{\ell}}/{\mathbb{F}}_{q}$. Then we can
specialize both ${\mathbf{a}}$ and ${\mathbf{b}}$ as $t\mapsto\tau$ (since
${\mathbf{a}}_{2}(\tau)$ and ${\mathbf{b}}_{2}(\tau)$ are both nonzero).

Let $\lambda_{0}\in\overline{\mathbb{F}_{q}}$. Then both ${\mathbf{a}}(\tau)$
and ${\mathbf{b}}(\tau)$ are torsion points for the Drinfeld module
$\Phi^{(\tau,\lambda_{0})}:{\mathbb{F}}_{q}[T]\longrightarrow{\rm
End}_{\overline{\mathbb{F}_{q}}}({\mathbb{G}}_{a})$ given by

$\Phi^{(\tau,\lambda_{0})}_{T}(x)=\tau x+\lambda_{0}x^{q}+x^{q^{r}}$

since each element in $\overline{\mathbb{F}_{q}}$ would be torsion for the
Drinfeld module $\Phi^{(\tau,\lambda_{0})}$ (which is itself defined over
$\overline{\mathbb{F}_{q}}$). Now, assume their respective orders $P(T)$ and
$Q(T)$ are both of degree at most $M$ (in $T$) for some real number $M>1$.

For each non-constant polynomial $R(T)\in{\mathbb{F}}_{q}[T]$, we let

$\displaystyle
g_{{\mathbf{a}},R(T)}(z):={\mathbf{a}}_{2}^{q^{r\deg_{T}R}}\cdot\Phi^{(z)}_{R(T)}({\mathbf{a}}),$
$\displaystyle
g_{{\mathbf{b}},R(T)}(z):={\mathbf{b}}_{2}^{q^{r\deg_{T}R}}\cdot\Phi^{(z)}_{R(T)}({\mathbf{b}}).$

Then

$g_{{\mathbf{a}},R(T)}(z),g_{{\mathbf{b}},R(T)}(z)\in{\mathbb{F}}_{q}[t][z]$

are polynomials of height at most $(2+D)^{q^{r\deg_{T}R}}$ (according to Lemma
2.7) and of degree $q^{r(\deg_{T}(R)-1)}$ (according to Lemma 2.5). Then let

$\displaystyle\tilde{g}_{{\mathbf{a}},M}(z):=\prod_{\begin{subarray}{c}1\leqslant\deg_{T}R\leqslant
M\\\ R\text{ is monic}\end{subarray}}g_{{\mathbf{a}},R(T)}(z)$
$\displaystyle\tilde{g}_{{\mathbf{b}},M}(z):=\prod_{\begin{subarray}{c}1\leqslant\deg_{T}R\leqslant
M\\\ R\text{ is monic}\end{subarray}}g_{{\mathbf{b}},R(T)}(z).$

The following two results are immediate consequences of Lemmas 2.5 and 2.7.

######  Lemma 3.1.

With the above notation, both $\tilde{g}_{{\mathbf{a}},M}(z)$ and
$\tilde{g}_{{\mathbf{b}},M}(z)$ have degree less than $q^{(M+1)(r+1)}$.

###### Proof.

For each $s=1,\ldots,M$, we have $q^{s}$ monic polynomials
$R(T)\in{\mathbb{F}}_{q}[T]$ of degree $s$. Since for each monic polynomial
$R(T)\in{\mathbb{F}}_{q}[T]$ of degree $s$, we have
$\deg_{z}(g_{{\mathbf{a}},R(T)}(z))=q^{r(s-1)}$ (according to Lemma 2.5), then
we obtain the desired conclusion from Lemma 3.1. ∎

######  Lemma 3.2.

With the above notation, we have that
${\mathfrak{h}}(\tilde{g}_{{\mathbf{a}},M}),{\mathfrak{h}}(\tilde{g}_{{\mathbf{b}},M})\leqslant
c_{2}(2+D)^{q^{(r+1)(M+1)}}$ for some constant $c_{2}$ depending only on $D$,
$q$ and $r$.

###### Proof.

Again using that we have $q^{s}$ monic polynomials of degree $s$ in
${\mathbb{F}}_{q}[T]$ along with Lemma 2.7, we derive that both
${\mathfrak{h}}(\tilde{g}_{{\mathbf{a}},M}(z))$ and
${\mathfrak{h}}(\tilde{g}_{{\mathbf{b}},M}(z))$ are bounded above by

$\sum_{s=1}^{M}q^{s}\cdot(2+D)^{q^{rs}}\ll(2+D)^{q^{(r+1)(M+1)}},$

where the implied constant depends only on $D$, $q$ and $r$. This concludes
our proof of Lemma 3.2. ∎

###  3.2. Concluding the proof

As previously noted, due to Theorem A (because ${\mathbf{a}}$ and
${\mathbf{b}}$ are ${\mathbb{F}}_{q}$-inearly independent), we know that there
exist at most $N_{0}$ solutions in $\overline{K}$ for the system:

$\tilde{g}_{{\mathbf{a}},M}(z)=\tilde{g}_{{\mathbf{b}},M}(z)=0.$

Then, by Lemma 2.4 (see also Lemmas 3.2 and 3.1), we know that there exists a
nonzero polynomial $\mathcal{T}\in{\mathbb{F}}_{q}[t]$ of degree at most

(3.1) $\left(2q^{(M+1)(r+1)}+1\right)\cdot
c_{2}(2+D)^{q^{(M+1)(r+1)}}<c_{3}(5+2D)^{q^{(M+1)(r+1)}},$

(for another constant $c_{3}$ depending only on $D$, $q$ and $r$) such that
for each $\tau\in\overline{\mathbb{F}_{q}}$ such that $\mathcal{T}(\tau)\neq
0$, then the specialization $t\mapsto\tau$ yields polynomials
$\tilde{g}_{{\mathbf{a}},M}(\tau;x),\tilde{g}_{{\mathbf{b}},M}(\tau;x)\in\overline{\mathbb{F}_{q}}[x]$
which have at most $N_{0}$ common roots $\lambda\in\overline{\mathbb{F}_{q}}$.
So, as long as $\tau$ generates a finite field extension
${\mathbb{F}}_{q^{\ell}}/{\mathbb{F}}_{q}$ of degree larger than
$\deg(\mathcal{T})$, then we are guaranteed that the system of equations

(3.2)
$\tilde{g}_{{\mathbf{a}},M}(\tau;\lambda)=\tilde{g}_{{\mathbf{b}},M}(\tau;\lambda)=0$

has at most $N_{0}$ solutions $\lambda\in\overline{\mathbb{F}_{q}}$. Hence, we
may take $M:=c_{0}\log\log(\ell)$ for some absolute constant $c_{0}>0$, which
depends only on $q$, $r$ and
$D=\max\\{\deg({\mathbf{a}}),\deg({\mathbf{b}})\\}$),so that $\ell$ is larger
than the bound from (3.1). Therefore, in this case, for each
$\lambda\in\overline{\mathbb{F}_{q}}$ other than the $N_{0}$ common solutions
to the system (3.2), we obtain that not both ${\mathbf{a}}(\tau)$ and
${\mathbf{b}}(\tau)$ are torsion points under the action of
$\Phi^{(\tau,\lambda)}$ of orders of degrees less than $M=c_{0}\log\log\ell$.
This concludes our proof of Theorem 1.3.

##  4\. Comments

In [GH13, Theorem 1.5], a slightly more general result has been proven than
the stated Theorem A. More precisely, the same conclusion as in Theorem A
holds if ${\mathbf{a}}$ and ${\mathbf{b}}$ are ${\mathbb{F}}_{q}$-linearly
independent elements of $\overline{\mathbb{F}_{q}}(t)$. The exact same proof
that we used for our Theorem 1.3 could be applied for
${\mathbf{a}},{\mathbf{b}}\in{\mathbb{F}}_{q^{s}}(t)$ (for some given
$s\in{\mathbb{N}}$); the only difference is that in Theorem A we would need to
replace ${\mathbb{F}}_{q}$ by ${\mathbb{F}}_{q^{s}}$ and
${\mathbb{F}}_{q^{\ell}}$ by ${\mathbb{F}}_{q^{s\ell}}$.

Furthermore, in [GH13, Theorem 1.4], a similar result as Theorem A is proven
for the same family of Drinfeld modules but replacing ${\mathbb{F}}_{q}(t)$
with any function field $K$ in which ${\mathbb{F}}_{q}$ is algebraically
closed. Once again, a similar result as the one from Theorem 1.3 can be
derived when starting with two ${\mathbb{F}}_{q}$-linearly independent points
${\mathbf{a}},{\mathbf{b}}\in K$. Indeed, for all but finitely many
specializations $t\mapsto\tau$ (where $\tau\in\overline{\mathbb{F}_{q}}$) we
have well defined specializations ${\mathbf{a}}\mapsto{\mathbf{a}}(\tau)$ and
${\mathbf{b}}\mapsto{\mathbf{b}}(\tau)$ and the rest of the argument follows
after replacing the technical statements from Subsection 2.3 with similar
results inferring heights (for more details regarding heights for Drinfeld
modules, see [Ghi07]). Now, in order to extend the result of our Theorem 1.3
to arbitrary ${\mathbb{F}}_{q}$-linearly independent points in
$\overline{{\mathbb{F}}_{q}(t)}$ (or more generally, to arbitrary one-
dimensional families of Drinfeld modules), the same strategy from this paper
applies as along as one would find a strengthening of [GH13, Theorems 1.4 and
1.5]; however, this last part seems very difficult at this moment since the
conclusions in [GH13, Theorems 1.4 and 1.5] are obtained through some
nontrivial arguments using valuation theory, which do not seem amenable to
generalizations.

Finally, one could consider a higher-dimensional variant of our Theorem 1.3 by
studying a family of Drinfeld modules parametrized by more than one variable.
So, for example, letting once again $K:={\mathbb{F}}_{q}(t)$, a natural
question would be to consider the $2$-dimensional family of Drinfeld modules
$\Psi^{(z_{1},z_{2})}:{\mathbb{F}}_{q}[T]\longrightarrow{\rm
End}_{K(z_{1},z_{2})}({\mathbb{G}}_{a})$ given by

$\Psi^{(z_{1},z_{2})}_{T}(x):=tx+z_{1}x^{q}+z_{2}x^{q^{2}}+x^{q^{3}}$

along with three ${\mathbb{F}}_{q}$-linearly independent points
${\mathbf{a}},{\mathbf{b}},{\mathbf{c}}\in K$. Then the _expectation_ is that
there are only finitely many pairs of $(z_{1},z_{2})\in\overline{K}^{2}$ such
that ${\mathbf{a}},{\mathbf{b}},{\mathbf{c}}$ are simultaneously torsion
points for the corresponding Drinfeld module $\Psi^{(z_{1},z_{2})}$. Even
though proving this expectation is beyond the known methods in arithmetic
dynamics, once such a result would be established, then our current proof
strategy would still apply verbatim to this more general setting using a
suitable multi-variables generalization of Lemma 2.4 (note that Lemmas 2.5 and
2.7 easily extend to higher-dimensional families of Drinfeld modules).

There are two possible approaches to such generalizing Lemma 2.4 to the
multivariate setting of polynomials
$f_{1},\ldots,f_{s}\in{\mathbb{F}}_{q}[t][\mathbf{x}]$ where
$\mathbf{x}=(x_{1},\ldots,x_{n})$. The first approach, which is perhaps rather
elaborate, is to follow the argument of [D’AOSS19] and establish an analogue
of [D’AOSS19, Theorem 2.1], in the function field case. In this case one can
use the machinery of heights in function fields, and for example, rely on the
effective Hilbert Nullstellensatz given by [DKS13, Theorem 5] (rather than on
its number field counterpart [DKS13, Theorem 5] as in [D’AOSS19]). The second
way is a little simpler and shorter, but may lead to weaker results. For this
we observe that if the system of equations

$f_{1}(\mathbf{x})=\ldots=f_{s}(\mathbf{x})=0$

has at most $N_{0}$ solutions in
$\mathbf{x}\in\overline{{\mathbb{F}}_{q}(t)}^{n}$, then the polynomial

$F\left(\mathbf{x}_{1},\ldots,\mathbf{x}_{N_{0}+1}\right):=\prod_{j=1}^{N_{0}}\left(x_{j,1}-x_{N_{0},1}\right)$

vanishes identically on the variety

(4.1) $f_{1}(\mathbf{x}_{j})=\ldots=f_{s}(\mathbf{x}_{j})=0,\qquad
j=1,\ldots,N_{0}+1.$

Hence the $s(N_{0}+1)+1$ polynomials $f_{i}(\mathbf{x}_{j})$, $i=1,\ldots s$,
$j=1,\ldots,N_{0}+1$ and
$1-zF\left(\mathbf{x}_{1},\ldots,\mathbf{x}_{N_{0}+1}\right)$ in
$n(N_{0}+1)+1$ variables do not have common zeros. Now we are under the
conditions of [DKS13, Theorem 5] which leads us to a multivariate analogue of
the relation (2.2) and thus allows us to estimate the number of
specialisations $t\mapsto\tau\in\overline{\mathbb{F}}_{q}$ for which the
corresponding polynomials
$f_{1}(\tau;\mathbf{x}),\ldots,f_{s}(\tau;\mathbf{x})$ have more that $N_{0}$
common zeros.

## Acknowledgements

The authors are grateful to Carlos D’Andrea and Martin Sombra for very useful
discussions and suggestions.

D. G. was partially supported by a Discovery Grant from NSERC, and I. S. was
partially supported by an ARC Grant DP200100355.

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# Accelerating Flow Simulations using Online Dynamic Mode Decomposition

Seung Won Suh
Department of Mechanical Science and Engineering
University of Illinois, Urbana-Champaign
Champaign, IL
<EMAIL_ADDRESS>
&Seung Whan Chung
Lawrence Livermore National Laboratory
Livermore, CA
<EMAIL_ADDRESS>
Peer-Timo Bremer
Lawrence Livermore National Laboratory
Livermore, CA
<EMAIL_ADDRESS>
&Youngsoo Choi
Lawrence Livermore National Laboratory
Livermore, CA
<EMAIL_ADDRESS>
###### Abstract

We develop an on-the-fly reduced-order model (ROM) integrated with a flow
simulation, gradually replacing a corresponding full-order model (FOM) of a
physics solver. Unlike offline methods requiring a separate FOM–only
simulation prior to model reduction, our approach constructs a ROM dynamically
during the simulation, replacing the FOM when deemed credible. Dynamic mode
decomposition (DMD) is employed for online ROM construction, with a single
snapshot vector used for rank-1 updates in each iteration. Demonstrated on a
flow over a cylinder with $\mathrm{Re}=100$, our hybrid FOM/ROM simulation is
verified in terms of the Strouhal number, resulting in a 4.4 times speedup
compared to the FOM solver.

##  1 Introduction

Iterative solvers for systems with a large number of degrees of freedom (DOFs)
can be found in almost all areas of computational science and engineering.
Physics simulations based on partial differential equations (PDEs) are one of
the classic examples, where the solution is represented with a finite number
of DOFs in space and marches in time as matrix equations are solved over
multiple time steps. Optimization is another example where one first solves a
primal problem to evaluate a quantity of interest and then solves its dual to
compute sensitivities iteratively for gradient descents, e.g., PDE-constrained
optimization or training of neural networks.

Although FOM–based solvers may generate predictions of high fidelity, they
easily get computationally infeasible if one tries to resolve a much wider
range of scales. At times when it is more important to get a rough estimate
rapidly, one might need to consider building ROMs which are less accurate but
cheaper than FOMs. Often used for building a ROM are modal analysis techniques
such as the DMD, which is a data-driven algorithm that extracts the dynamics
of the temporal evolution of a system [1].

Here we present an efficient workflow for constructing the ROM for flow
simulations. Before getting started, we emphasize that such traditional
reduced-order modeling is not different from the techniques used to reduce
dimensionalities in the field of machine learning. For instance, the principal
component analysis (PCA) is merely another name of the proper orthogonal
decomposition (POD) [2], which is widely used for reduced-order modeling in
physics simulations [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16].
Therefore, our work of reducing the size of flow simulations using the DMD can
be regarded adjacent to machine learning–based surrogate modeling and the
workflow that we are presenting is not limited to the DMD but can be applied
to any type of technique for dimensionality reduction.

A plethora of works have already shown the capability of the DMD to learn the
nonlinear dynamics including that of fluid flows [17], pore-collapse [18], and
radiative diffusion problem [19]. However, most of the studies are limited to
offline training. That is, many full-order model simulations need to be run to
generate the whole set of samples, and then the ROM is trained separately from
the simulation using those samples. This may be undesirable when the
dimensionality of the underlying parameter space is high, leading to the curse
of dimensionality and requiring a large number of sampled training points. In
such scenarios, efficiency can be enhanced by adopting an on-the-fly
approach—interrupting the full-order model simulation and substituting the
reduced-order model (ROM) at the relevant time step—rather than adhering to
separate offline training and online inference steps. Via such a hybrid
FOM/ROM method, we anticipate achieving two things at once: (1) accelerating
the simulation itself, and (2) constructing a ROM that can be reused later for
other simulations having similar setups, e.g., those for parametric studies.
The existing on-the-fly data-driven approach includes projection-based reduced
order modeling on PDE-constrained optimization [20, 21]. Similar approaches of
using online POD have also been introduced by [22, 23], as well as those using
artificial neural networks by [24] and [25].

In this work, we demonstrate a workflow for hybrid FOM/DMD flow
simulations—which has not yet been attempted up to the authors’
knowledge—where the FOM is gradually replaced by the DMD trained on the fly.
Each iteration consists of two key steps: (1) to update the DMD and (2) to
replace the FOM with the DMD if possible. In addition, if the DMD is thought
to have been converged, we end the simulation and directly make a prediction
at the final time step, skipping all the remaining time steps and thereby
significantly reducing the cost. This hybrid method has been applied to the
simulation of a flow over a cylinder, followed by a comparison to a FOM–only
simulation.

##  2 Methodology

###  2.1 Workflow

Figure 1: Workflow of the hybrid FOM/DMD simulation of a dynamical system.

Figure 1 describes the workflow of a hybrid FOM/DMD simulation. Like standard
solvers for time-dependent systems, we start the time integration with an
initial condition. At the beginning of the first time step, there is no DMD
initialized so we call the FOM and solve the system. Once we get a new
snapshot of the solution, we use it to initialize the DMD, and then proceed to
the next time step. At the beginning of the second time step, we first make a
prediction on the solution at the second time step with our initialized DMD.
If it is accurate enough, i.e. the residual of the prediction is smaller than
the tolerance, we use that prediction as our new snapshot and move on to the
next time step without calling the FOM. If not, we call the FOM and use the
FOM prediction as the new snapshot to update the previously-trained DMD model
in a rank-1 update fashion. While the DMD gradually replaces the FOM, we also
check whether the DMD itself has converged and if so, we exit the loop and go
directly to $t_{\mathrm{final}}$ to make a prediction.

###  2.2 Error Estimator

One way to evaluate the accuracy of the prediction at the $k$-th time step
made by the constructed DMD is to measure the local truncation error
$\boldsymbol{e}_{\mathrm{DMD}}^{k}=\boldsymbol{u}_{\mathrm{DMD}}^{k}-\boldsymbol{u}_{\mathrm{FOM}}^{k}$
where $\boldsymbol{u}^{k}$ is the prediction of the solution at the $k$-th
time step by either the DMD or the FOM. However, computing the error should be
avoided since it requires solving the FOM for
$\boldsymbol{u}_{\mathrm{FOM}}^{k}$. Instead, we compute the residual as a
representative of the local error. If a FOM solves a linear system
$\boldsymbol{A}\boldsymbol{u}=\boldsymbol{b}$, the residual is defined as
$\boldsymbol{r}(\boldsymbol{u})=\boldsymbol{b}-\boldsymbol{A}\boldsymbol{u}$
which does not require solving the matrix equation. As
$\boldsymbol{r}(\boldsymbol{u}_{\mathrm{DMD}})=-\boldsymbol{A}\boldsymbol{e}_{\mathrm{DMD}}$,
the residual is a linear map of the error and is a valid estimator of the
error. Likewise, for nonlinear systems, one can derive residual-based error
bounds, indicating that the residual norm is a good error indicator. In a
nutshell, whenever we make a prediction with the DMD, we compute the residual
as $\boldsymbol{r}(\boldsymbol{u}_{\mathrm{DMD}})$ and if its norm is smaller
than a certain level of tolerance $\epsilon_{\mathrm{DMD}}$, we trust the DMD
and use its prediction as our new snapshot. Note that the residual depends on
the numerical scheme used for the FOM.

###  2.3 Online DMD

The DMD approximates either linear or nonlinear dynamics using a linear
operator. Consider a dynamical system that evolves over time as
$\boldsymbol{u}^{k+1}=\boldsymbol{F}(\boldsymbol{u}^{k})$ where
$\boldsymbol{u}^{k}\in\mathbb{R}^{N}$ is an $N$-dimensional state vector at
time steps $k=0,1,\cdots,n$ and
$\boldsymbol{F}:\mathbb{R}^{N}\to\mathbb{R}^{N}$ is an operator that describes
the temporal evolution of the state. Then we can construct two matrices
composed of snapshot vectors
$\boldsymbol{U}^{+}=\left[\boldsymbol{u}^{n}|\boldsymbol{u}^{n-1}|\cdots|\boldsymbol{u}^{1}\right]\in\mathbb{R}^{N\times
n}$ and
$\boldsymbol{U}^{-}=\left[\boldsymbol{u}^{n-1}|\boldsymbol{u}^{n-2}|\cdots|\boldsymbol{u}^{0}\right]\in\mathbb{R}^{N\times
n}.$ The DMD matrix is then given as
$\boldsymbol{A}=\boldsymbol{U}^{+}(\boldsymbol{U}^{-})^{\dagger}\approx\boldsymbol{U}^{+}\boldsymbol{R}\boldsymbol{S}^{-1}\boldsymbol{L}^{*}=\boldsymbol{A}_{r}\in\mathbb{R}^{N\times
N}$ where $\boldsymbol{U}^{-}=\boldsymbol{L}\boldsymbol{S}\boldsymbol{R}^{*}$
is the truncated SVD at rank $r\leq n$ ($\boldsymbol{L}\in\mathbb{R}^{N\times
r},\boldsymbol{S}\in\mathbb{R}^{r\times
r},\boldsymbol{R}\in\mathbb{R}^{m\times r}$) and † is the Moore-Penrose
inverse of a matrix. Furthermore, the full-rank reduced DMD matrix is defined
as
$\tilde{\boldsymbol{A}}_{r}=\boldsymbol{L}^{*}\boldsymbol{U}^{+}\boldsymbol{R}\boldsymbol{S}^{-1}\in\mathbb{R}^{r\times
r}$ and the projected DMD modes are
$\boldsymbol{\Phi}=\boldsymbol{L}\boldsymbol{V}\in\mathbb{R}^{N\times r}$
where
$\tilde{\boldsymbol{A}}_{r}=\boldsymbol{V}\boldsymbol{\Lambda}\boldsymbol{V}^{-1}$
is the eigenvalue-decomposition of $\tilde{\boldsymbol{A}}_{r}$. Once the DMD
matrices are constructed, one can predict the solution at any time $t$ based
on the analytical form
$\boldsymbol{u}(t)=\boldsymbol{\Phi}\boldsymbol{\Lambda}^{t/\Delta
t}\boldsymbol{\Phi}^{\dagger}\boldsymbol{u}(0).$ Hence, the DMD forms a ROM
that makes a prediction at a constant time complexity of $\mathcal{O}(Nr)$,
independent of $t$.

In our hybrid approach, updating the DMD matrices by a single rank is
sufficient as only a single snapshot is appended to the snapshot matrix every
iteration. Incremental SVD by [26] is an algorithm that efficiently makes a
rank-1 update to the SVD of a matrix. The essence of the algorithm is to avoid
costly matrix-matrix multiplications and utilize matrix-vector multiplications
to get a speedup by $\mathcal{O}(r)$. We can employ the ideas from incremental
SVD to efficiently make rank-1 updates on the DMD matrices. Although several
authors have already introduced the idea of online DMD [27, 28], ours differ
from theirs as [28] is based on the Sherman–Morrison formula and [27] used the
modified version of the original algorithm [26], while we stick to the
original version to update the DMD matrices. In the end, each DMD update costs
$\mathcal{O}(Nr)$, which is more efficient than naively constructing the
matrices from scratch that costs $\mathcal{O}(Nnr)$ for $N\gg n\gg r$. Details
of the implementation are not presented in this paper due to the limited
space.

##  3 Numerical Experiments & Results

The prescribed hybrid workflow was applied to simulate a three-dimensional
incompressible flow over a cylinder. The cylinder has a diameter of $D$ with
its axis along the $z$-direction at $(x_{c},y_{c})=(5D,0)$. The computational
domain was defined as
$\Omega=[0,L_{x}]\times[-L_{y}/2,L_{y}/2]\times[-L_{z}/2,L_{z}/2]$ with the
dimensions $(L_{x},L_{y},L_{z})=(25D,4.1D,4.1D)$. Dirichlet boundary
conditions were set for the velocities at the inflow ($x=0$) and the top and
bottom walls ($y=\pm L_{y}/2$) as $(u,v)=(U,0)$, and pure Neumann boundary
conditions were set elsewhere. For the pressure, pure Neumann boundary
conditions were applied at the inflow and the top and bottom walls, and the
Dirichlet boundary condition was used at the outflow $(x=L_{x})$ as $p=0$. The
kinematic viscosity $\nu$ was set to match the Reynolds number
$\mathrm{Re}=UD/\nu=100$ where we can anticipate mild vortex streets. Although
the top and bottom boundaries were quite close to the cylinder, we were able
to observe a distinct vortex-shedding structure throughout the simulation.

An incompressible flow solver in MFEM [29] was used as the FOM, which uses a
nodal $H^{1}$ finite element discretization with polynomial order of $3$ in
space and an implicit (of order 3) / explicit (of order 3) time integration of
the viscous and convective terms in the momentum equation, respectively. More
detailed explanations of the numerical scheme can be found in [29, 30, 31].

An unstructured coarse baseline mesh was further refined within the solver
where the total number of DOFs was 585,732. A preliminary high-fidelity FOM
simulation was first run for $t^{*}=Ut/D=50$ starting from the zero flow until
it was thought to have entered the limit cycle of the vortex shedding. The
time step size was adjusted so the maximum CFL number remained around 0.2
throughout the simulation. Then, the final snapshot of the solution was used
to initialize flow fields for the hybrid and restarted FOM–only simulations,
which were both run until $t^{*}=100$. For a fair comparison, the level of
fidelity of the restarted FOM–only simulation has been lowered down to that of
the DMD: the tolerance level of the conjugate-gradient solver
($\epsilon_{\mathrm{CG}}$) for the FOM–only simulation and the tolerance level
for the accuracy of the DMD ($\epsilon_{\mathrm{DMD}}$) were both set to
$10^{-5}$. Note that the FOM for the hybrid simulation retains the high
fidelity of $\epsilon_{\mathrm{CG}}=10^{-12}$.

|
---|---
Figure 2: Velocity magnitude plotted on a 2-D slice normal to the $z$-axis at
$t^{*}=50$ (left); temporal spectrum of $u_{y}$ sampled downstream.

Temporal spectra of the $y$-velocity at a single point $(x,y,z)=(10D,0,0)$
from the hybrid and a high-fidelity ($\epsilon_{\mathrm{CG}}=10^{-12}$)
FOM–only simulations were compared for verification. Frequencies of the vortex
shedding predicted by the two simulations were identical within the precision
of the discrete values of frequency. The Strouhal number
$\mathrm{St}=fD/U$—the dimensionless frequency of the vortex shedding—was 0.22
in both simulations, which is around the reported value of 0.16 at
$\mathrm{Re}=100$ [32]. The discrepancy is thought to be due to the boundary
effect.

Figure 3: Wall time per time step: hybrid vs. FOM (top); history of the
relative residual of $u_{\mathrm{DMD}}$ and the rank of $\boldsymbol{A}_{r}$
(bottom). The horizontal axis only shows until the 1,600${}^{\text{th}}$ time
step out of 10,000 time steps of the FOM–only simulation.

Figure 3 shows the cost of the two simulations and the history of the residual
of $u$ from the hybrid simulation. Notice that at the early stage of the
simulation, the residual keeps going up and down around the level of tolerance
($10^{-5}$) so the FOM is frequently called. As the simulation goes on, the
model gets more accurate and hits below the level of tolerance more often,
leading to less frequent calls of the FOM. This depicts how the FOM is
gradually replaced by the DMD throughout the simulation as we expected. In
total, the hybrid and the low-fidelity FOM–only simulations took 375s and
1648s for $10^{4}$ time steps, respectively, thereby showing about a 4.4-times
speedup. Moreover, even when we choose not to skip all the remaining time
steps of the simulation, a single time-stepping using the converged DMD is
5.77 times faster than using the low-fidelity FOM so the total cost of our
approach will remain cheaper.

| Hybrid (FOM/DMD) | Low-fidelity FOM | Speedup
---|---|---|---
Total simulation time (s) | 375 | 1648 | 4.39
Single-$\Delta t$ prediction time (s) | 0.0286 | 0.165 | 5.77

Table 1: Two different types of speedups, in terms of the total simulation
time and the cost of a single time-stepping. The total simulation time for the
FOM/DMD approach includes the time for making a prediction, checking accuracy,
and training. The cost of a single time-stepping only measures the time spent
on making a prediction, based on the converged DMD.

##  4 Conclusion

We used the DMD to reduce the dimension of the FOM that is evaluated
iteratively during flow simulations, where the DMD is constructed on the fly
and gradually replaces the FOM. This has been demonstrated on a simulation of
a flow over a cylinder, and we were able to gain a 4.4-times speedup.

The fact that the DMD discovers underlying physics in a data-driven way
without any prior knowledge of the governing equations resembles the way
machine learning is used for scientific discovery. Therefore, our work has
both aspects of traditional reduced-order modeling and machine learning and
will be a good resource for following studies on model reduction of dynamical
systems in both fields.

## Acknowledgments and Disclosure of Funding

This work was performed under the auspices of the U.S. Department of Energy by
Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344 and
was supported by Laboratory Directed Research and Development funding under
projects 22-SI-004 and 22-SI-006. Y. Choi was supported for this work by the
U.S. Department of Energy, Office of Science, Office of Advanced Scientific
Computing Research, as part of the CHaRMNET Mathematical Multifaceted
Integrated Capability Center (MMICC) program, under Award Number DE-SC0023164.
IM release: LLNL-CONF-855140

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