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1 Introduction
The concept of an exceptional unit (exunit, for short), defined as a unit u in a ring R such that
$1-u$
is also a unit, was first introduced by Nagell [Reference Nagell11] in 1969 to study cubic Diophantine equations. Since then, it has found profound applications in algebraic number theory, most notably in Lenstra’s work on Euclidean number fields [Reference Lenstra8]. Apart from these applications, the counting function of exunits itself possesses a rich arithmetic structure. Harrington and Jones [Reference Harrington and Jones6] investigated the dynamical behaviour of the iterations of this counting function over
$\mathbb {Z}/n\mathbb {Z}$
.
Counting the number of exunits satisfying certain constraints is a focus of many papers. Sander [Reference Sander13] obtained a formula for the number of solutions in
$\mathbb {Z}/n\mathbb {Z}$
to the linear congruence
$x_1+x_2\equiv c\pmod {n}$
with both
$x_1$
and
$x_2$
being exunits. This was later extended to the sums of k exunits by Yang and Zhao [Reference Yang and Zhao14], and further generalised to arbitrary finite commutative rings by Miguel [Reference Miguel9]. Beyond linear sums, there have been attempts to address nonlinear constraints. Mollahajiaghaei [Reference Mollahajiaghaei10] derived a formula for the number of solutions to the quadratic congruence
$x_1^2+x_2^2+\cdots +x_k^2\equiv c\pmod {n}$
with each
$x_i$
being an exunit, and Feng and Hong [Reference Feng and Hong4] investigated
$x_1^e+x_2^e+\cdots +x_k^e\equiv c\pmod {n}$
with a given positive integer e.
Anand et al. [Reference Chattopadhyay and Roy1] generalised this notion to f-exunits. For a polynomial
$f(X)\in \mathbb {Z}[X]$
and an integer
$n\geq 2$
, an element
$a\in \mathbb {Z}/n\mathbb {Z}$
is said to be an f-exunit if
$\mathrm {gcd}(f(a),n)=1$
. This generalisation naturally prompted investigations into the counting functions of f-exunits similar to those for classical exunits (see [Reference Chattopadhyay and Roy1–Reference Feng and Hong3, Reference Zhao, Hong and Zhu15] for results on sumsets of f-exunits).
Previous studies have primarily focused on the ring of rational integers
$\mathbb {Z}$
or abstract finite rings, with investigations largely restricted to linear constraints (that is, sumsets). In his study of finite rings, Miguel [Reference Miguel9] explicitly pointed out that these counting problems can be extended to the ring of integers of an algebraic number field. Another natural and significant question is: instead of a single linear equation, can we determine the counting formula of f-exunits for a general polynomial equation or algebraic varieties defined by systems of polynomial equations?
In this paper, we consider such results in the setting of number fields. Our main result (Theorem 1.2) provides a counting formula for f-exunits lying on a variety X modulo an ideal
$\mathfrak {n}$
.
To state our main results precisely, we first generalise the definition of f-exunits to the setting of an algebraic number field.
Definition 1.1. Let K be an algebraic number field with
$\mathcal {O}_K$
its ring of integers and
$\mathfrak {n}$
a nonzero integral ideal of
$\mathcal {O}_K$
. For a nonconstant polynomial
$f(X)\in \mathcal {O}_K[X]$
, an element
$\alpha \in \mathcal {O}_K/\mathfrak {n}$
is called an f-exunit modulo
$\mathfrak {n}$
or an
$(f,\mathfrak {n})$
-exunit if
$f(\alpha )$
is a unit in the ring
$\mathcal {O}_K/\mathfrak {n}$
. The set of all
$(f,\mathfrak {n})$
-exunits in
$\mathcal {O}_K/\mathfrak {n}$
is denoted by
$\mathscr {E}_{f,\mathfrak {n}}$
.
With the definition of f-exunits extended to the ring of integers, we now turn to the counting problem for solutions satisfying a system of polynomial equations. We use the language of schemes, which simplifies the statement and proof of our main theorem.
For a closed subscheme X of
$\mathbb {A}^n_{\mathcal {O}_K}$
(the n-dimensional affine space over
$\mathcal {O}_K$
), we define the set
where
$X(\mathcal {O}_K/\mathfrak {n})$
denotes the
$\mathcal {O}_K/\mathfrak {n}$
-points of X.
Our main result provides an explicit counting formula for the cardinality of this set when the closed subscheme X satisfies smoothness conditions.
Theorem 1.2. Given a number field K with ring of integers
$\mathcal {O}_K$
, let
$f\in \mathcal {O}_K[x]$
be a polynomial and let X be a smooth closed subscheme of
$\mathbb {A}^n_{\mathcal {O}_K}$
with
$\mathrm {codim}(X)=d$
. For a nonzero integral ideal
$\mathfrak {n}$
of
$\mathcal {O}_K$
such that X has a good reduction at
$\mathfrak {p}$
for every prime ideal
$\mathfrak {p}\mid \mathfrak {n}$
,
where
$\mathcal {N}^f(\mathfrak {p},X)=\{(x_1,\ldots ,x_n)\in X(\mathcal {O}_K/\mathfrak {p}):\prod _{i=1}^{n}\overline {f}(x_i)=0\}$
,
$\mathrm {N}_{K/\mathbb {Q}}$
denotes the norm map in
$K/\mathbb {Q}$
and
$\overline {f}$
is the image of f under the projection
$\mathcal {O}_K\to \mathcal {O}_K/\mathfrak {p}$
.
While Theorem 1.2 establishes a formula for
$\#\mathcal {E}_{f,\mathfrak {n}}(X)$
, it relies on the local point counts
$\#X(\mathcal {O}_K/\mathfrak {p})$
and the intersection sizes
$\#\mathcal {N}^f(\mathfrak {p},X)$
, which vary irregularly with the prime
$\mathfrak {p}$
. Using the Lang–Weil inequality, we can derive a bound for
$\#\mathcal {E}_{f,\mathfrak {n}}(X)$
that depends only on the geometric invariants of X and the absolute norm of the ideal
$\mathfrak {n}$
.
Theorem 1.3. Given a number field K with ring of integers
$\mathcal {O}_K$
, let
$f\in \mathcal {O}_K[x]$
be a polynomial and let X be a smooth closed subvariety of
$\mathbb {A}^n_{\mathcal {O}_K}$
with
$\mathrm {codim}(X)=d$
, which is geometrically irreducible over K. For a nonzero integral ideal
$\mathfrak {n}$
of
$\mathcal {O}_K$
such that X has a good reduction at
$\mathfrak {p}$
for any prime ideal
$\mathfrak {p}\mid \mathfrak {n}$
,
The implied constant in the O-term depends only on
$K,X$
and f.
The error term in Theorem 1.3 that provides an asymptotic bound for general smooth varieties can be significantly improved when the variety X has low degree. Since previous studies have focused mostly on linear and quadratic congruences, as discussed before, we derive the following sharper estimate for varieties of degree no more than 2.
Theorem 1.4. Given a number field K with ring of integers
$\mathcal {O}_K$
, let
$f\in \mathcal {O}_K[x]$
be a polynomial and let X be a smooth closed subvariety of
$\mathbb {A}^n_{\mathcal {O}_K}$
with
$\mathrm {codim}(X)=d$
and
$\mathrm {deg}X\le 2$
, which is geometrically irreducible over K. For a nonzero integral ideal
$\mathfrak {n}$
of
$\mathcal {O}_K$
such that X has a good reduction at
$\mathfrak {p}$
for any prime ideal
$\mathfrak {p}\mid \mathfrak {n}$
,
The implied constant in the O-term depends only on
$K,X$
and f.
Our approach relies on the multiplicative structure of the counting function derived from the Chinese remainder theorem in Dedekind domains. For the counting functions in the local case, we employ a Hensel-type technique to lift solutions from the residue field
$\mathcal {O}_K/\mathfrak {p}$
to
$\mathcal {O}_K/\mathfrak {p}^k$
. To establish the asymptotic estimate for
$\#\mathscr {E}_{f,\mathfrak {n}}(X)$
, we use the well-known Lang–Weil inequality to control the error terms arising from the local points on the variety.
In Section 2, we provide the proof of Theorem 1.2. The argument begins with establishing the multiplicativity of the counting function
$\#\mathscr {E}_{f,\mathfrak {n}}(X)$
via the Chinese remainder theorem. Then, we derive the local counting formulae by using Hensel lifting. To illustrate the concrete application of our counting formula, we present a detailed example involving a variety X over the quadratic field
$\mathbb {Q}(\sqrt {-5})$
at the end of this section.
In Section 3, we focus on the asymptotic behaviour of
$\#\mathscr {E}_{f,\mathfrak {n}}(X)$
as the norm of the ideal
$N(\mathfrak {n})$
tends to infinity. Using the Lang–Weil inequality, we derive a general estimate for the number of
$(f,\mathfrak {n})$
-exunits, which depends on the dimension and degree of the variety, and the absolute norm of
$\mathfrak {n}$
. Furthermore, we investigate the specific case where
$\deg X \le 2$
. Noting that the leading error term coefficient vanishes for quadratic varieties and linear varieties, we obtain a refined estimate that improves the error bound from
$\exp ({O({\log \mathrm {N}_{K/\mathbb {Q}}(\mathfrak {n})}/{\log \log \mathrm {N}_{K/\mathbb {Q}}(\mathfrak {n})})})$
to
$\exp (O(\log \log \mathrm {N}_{K/\mathbb {Q}}(\mathfrak {n})))=(\log \mathrm {N}_{K/\mathbb {Q}}(\mathfrak {n}))^{O(1)}$
.
2 Proof of Theorem 1.2
In this section, we complete the proof of Theorem 1.2 by employing the Euler decomposition and the Hensel lifting technique. Moreover, we present a concrete example to illustrate the application of our counting formula.
For any
$f \in \mathcal {O}_K[x_1,\ldots ,x_m]$
, we denote by
$\overline {f} \in (\mathcal {O}_K/\mathfrak {n})[x_1,\ldots , x_m]$
its image under the reduction map modulo
$\mathfrak {n}$
. For brevity, we shall write
$f(\alpha )$
to denote
$\overline {f}(\alpha )$
for
${\alpha \in (\mathcal {O}_K/\mathfrak {n})^m}$
.
First, we prove that
$\#\mathscr {E}_{f,\mathfrak {n}}(X)$
is a multiplicative function with respect to the ideals of
$\mathcal {O}_K$
.
Lemma 2.1. Let
$\mathfrak {m},\mathfrak {n}$
be integral ideals of
$\mathcal {O}_K$
satisfying
$\mathfrak {m}+\mathfrak {n}=\mathcal {O}_K$
and let X be a subscheme of
$\mathbb {A}_{\mathcal {O}_K}^n$
. Then,
Proof. According to the Chinese remainder theorem, we have a canonical ring isomorphism
Since the functor of points of a scheme is compatible with finite products of rings, for any subscheme of
$\mathbb {A}^n_{\mathcal {O}_K}$
, we have the bijection
Under this identification, the condition that
$f(x_i)$
is a unit in
$\mathcal O_K/\mathfrak {mn}$
is equivalent to the condition that its images in both
$\mathcal O_K/\mathfrak m$
and
$\mathcal O_K/\mathfrak n$
are units. Therefore, the above bijection restricts to the bijection
which proves the lemma.
Remark 2.2. We note that the smoothness and good reduction assumptions imposed in Theorem 1.2 are not required for this lemma. The multiplication property established here is a purely ring-theoretic consequence of the Chinese remainder theorem.
Let
$\mathfrak {n}=\mathfrak {p}_1^{\alpha _1}\cdots \mathfrak {p}_r^{\alpha _r}$
be an ideal of
$\mathcal {O}_K$
with
$\mathfrak {p}_1,\ldots ,\mathfrak {p}_r$
distinct prime ideals of
$\mathcal {O}_K$
. By Lemma 2.1,
$\mathscr {E}_{f,\mathfrak {n}}(X)\cong \prod _{i=1}^r \mathscr {E}_{f,\mathfrak {p}_i^{\alpha _i}}(X)$
. The following lemma addresses the lifting problem in the case where
$\mathfrak {n}$
is a power of a prime ideal.
Lemma 2.3. Let X be a smooth closed subscheme of
$\mathbb {A}^n_{\mathcal {O}_K}$
with
$\mathrm {codim}\,X=d$
and let
$\mathfrak {p}$
be a prime ideal of
$\mathcal {O}_K$
such that X has a good reduction at
$\mathfrak {p}$
. For
$\mathbf {a}\in X(\mathcal {O}_K/\mathfrak {p}^k)$
, the set
has cardinality
$(\mathrm {N}_{K/\mathbb {Q}}(\mathfrak {p}))^{n-d}$
.
Proof. Let
$\hat {\mathcal {O}}_K$
be the completion of
$\mathcal {O}_K$
at
$\mathfrak {p}$
. Then,
$\hat {\mathcal {O}}_K$
is a discrete valuation ring with uniformiser
$\pi $
. We have
$\hat {\mathcal {O}}_K/(\pi ^m\hat {\mathcal {O}}_K)=\mathcal {O}_K/\mathfrak {p}^m$
for all positive integers m. Hence, all counting statements modulo
$\mathfrak {p^m}$
can be carried out after replacing
$\mathcal {O}_K$
by its
$\mathfrak {p}$
-adic completion. Without loss of generality, we may assume that
$\mathcal {O}_K$
is complete at
$\mathfrak {p}$
and, hence, a discrete valuation ring. Let
$\pi $
denote the uniformiser of
$\mathcal {O}_K$
, that is,
$(\pi )=\mathfrak {p}$
.
Let I denote the ideal of X. Since
$O_K[x_1,\ldots ,x_n]$
is Noetherian, the ideal I is finitely generated. Let
$\{g_1,\ldots ,g_m\}$
be its set of generators.
All points of
$X(\mathcal {O}_K/\mathfrak {p}^{k+1})$
lifting
$\mathbf {a}$
have the form
$\boldsymbol {\alpha } \equiv \mathbf {a}+\boldsymbol {\delta }\pi ^k\pmod {\pi ^{k+1}}$
for some
$\boldsymbol {\delta } \in ({\mathcal {O}}_K/(\pi ))^n$
because
$\pi ^k\boldsymbol {\delta } $
is well defined modulo
$\pi ^{k+1}$
.
Let
$\mathbf {g}$
denote the polynomial vector
$(g_1,\ldots ,g_m)\in (\mathcal {O}_K[x_1,\ldots ,x_n])^m$
. Consider the equation
where
$\mathrm {J}(X)$
denotes the Jacobi matrix of X. Since
$\mathbf {g}(\mathbf {a})\equiv 0\pmod {\pi ^k}$
, we may write
$\mathbf {g}(\mathbf {a})=\pi ^k \mathbf {a}^{\prime }$
for some
$\mathbf {a}^\prime \in (\mathcal {O}_K/(\pi ))^n$
. The equation above can be simplified to
Since X is smooth with
$\mathrm {codim}\,X=d$
and has a good reduction at
$\mathfrak {p}$
, we have
$\mathrm {rank}(\mathrm {J}(X))=d$
over the residue field
$\mathcal {O}_K/\mathfrak {p}$
. Thus, there are
$(\#({\mathcal {O}}_K/\mathfrak {p}))^{n-d}$
distinct solutions. Hence, the cardinality of the set
is equal to
$(\#({\mathcal {O}}_K/\pi ))^{n-d}=(\#({\mathcal {O}}_K/\mathfrak {p}))^{n-d}=(\mathrm {N}_{K/\mathbb {Q}}(\mathfrak {p}))^{n-d}$
.
We can now derive the explicit counting formula for prime powers by inductively applying the lifting property established in Lemma 2.3.
Proposition 2.4. Let X be a smooth closed subscheme of
$\mathbb {A}^n_{\mathcal {O}_K}$
with
$\mathrm {codim}\,X=d$
and let
$\mathfrak {p}$
be a prime ideal of
$\mathcal {O}_K$
such that X has a good reduction at
$\mathfrak {p}$
. For a positive integer e,
Proof. Consider the exact sequence
The projection
$\varphi $
induces a map
Since an element of
$\mathcal O_K/\mathfrak p^e$
is a unit if and only if its image in
$\mathcal O_K/\mathfrak p$
is a unit, the map
$\phi $
is well defined.
Recall that X is smooth with
$\mathrm {codim}\,X=d$
and has a good reduction at
$\mathfrak {p}$
. By applying Lemma 2.3 inductively, the cardinality of the fibre of the projection
$X(\mathcal O_K/\mathfrak p^e)\to X(\mathcal O_K/\mathfrak p)$
over any
$\mathbf a\in X(\mathcal O_K/\mathfrak p)$
is equal to
$(\mathrm N_{K/\mathbb Q}(\mathfrak p))^{(n-d)(e-1)}$
.
For any
$\mathbf {a}\in X(\mathcal {O}_K/\mathfrak {p})$
, there exist
$(\mathrm {N}_{K/\mathbb {Q}}(\mathfrak {p}))^{(n-d)(e-1)}$
elements lifting
$\mathbf {a}$
in
$X(\mathcal {O}_K/\mathfrak {p}^e)$
. Since the unit condition is preserved under reduction modulo
$\mathfrak p$
, each element of
$\mathscr E_{f,\mathfrak p}(X)$
admits exactly
$(\mathrm N_{K/\mathbb Q}(\mathfrak p))^{(n-d)(e-1)}$
lifts in
$\mathscr E_{f,\mathfrak p^e}(X)$
. Hence,
Since
$\#\mathscr {E}_{f,\mathfrak {p}}(X)=\#X(\mathcal {O}_K/\mathfrak {p})-\# \mathcal {N}^f(\mathfrak {p},X)$
, we have
This completes the proof of Proposition 2.4.
With the explicit counting formula for prime powers established in Proposition 2.4, we are ready to prove our main result.
Proof of Theorem 1.2
Let
$\mathfrak {n}=\mathfrak {p}_1^{e_1}\cdots \mathfrak {p}_g^{e_g}$
be the prime ideal factorisation of
$\mathfrak {n}$
, where
$\mathfrak {p}_1,\ldots ,\mathfrak {p}_g$
are distinct prime ideals of
$\mathcal {O}_K$
. According to Lemma 2.1,
Using Proposition 2.4,
This completes the proof of Theorem 1.2.
Having established the general counting formula, we now present a concrete example to show its application. In the following example, we explicitly compute the various local quantities involved and conclude with the precise count of f-exunits modulo
$\mathfrak {n}$
for a specific variety.
Example 2.5. Let
$K=\mathbb {Q}(\sqrt {-5})$
, so that its ring of integers is
$\mathcal {O}_K=\mathbb {Z}[\sqrt {-5}]$
. We consider the variety
$X\subset \mathbb {A}^2_{\mathcal {O}_K}$
defined by the circle equation
$x^2+y^2=c$
, where the parameter
$c\in \mathbb {Z}\setminus \{0\}$
. This variety X is smooth and geometrically irreducible. Let
$f(x)=x-a$
be a polynomial in
$ \mathbb {Z}[x]$
.
To ensure good reductions, we assume the ideal
$\mathfrak {n}$
is coprime to
$(2c)$
. Let
$\mathfrak {p}$
be a prime ideal of
$\mathcal {O}_K$
lying above a prime
$p\in \mathbb {Z}$
. The number of rational points on X over the residue field
$\mathcal {O}_K/\mathfrak {p}$
is given by the well-known formula
where
$({\cdot }/{\mathfrak {p}})$
is the generalised Legendre symbol.
Next, we calculate the intersection number
$\#\mathcal {N}^f(\mathfrak {p},X)$
. The set
$\mathcal {N}^f(\mathfrak {p},X)$
consists of the points in
$X(\mathcal {O}_K/\mathfrak {p})$
satisfying
$x=a$
or
$y=a$
. If
$c-a^2$
is not a square in
$\mathcal {O}_K/\mathfrak {p}$
, then
$\#\mathcal {N}^f(\mathfrak {p},X)=0$
. If
$c-a^2$
is a square in
$\mathcal {O}_K/\mathfrak {p}$
, then there exists
$b\in \mathcal {O}_K$
such that
$b^2\equiv c-a^2\pmod {\mathfrak {p}}$
. The intersection set
$\mathcal {N}^f(\mathfrak {p},X)$
contains at most four points:
$(a,b),(a,-b),(b,a)$
and
$(-b,a)$
.
The exact cardinality depends on the overlapping of these points. If
$a^2-c\equiv 0$
$\pmod {\mathfrak {p}}$
, then
$b=-b$
, implying that there are
$2$
points in
$\mathcal {N}^f(\mathfrak {p},X)$
. If
$2a^2-c\equiv 0$
$\pmod {\mathfrak {p}}$
, then
$b=\pm a$
, implying that there are
$3$
points in
$\mathcal {N}^f(\mathfrak {p},X)$
. Otherwise, there are exactly 4 distinct points in
$\mathcal {N}^f(\mathfrak {p},X)$
. Note that for any rational integer r, we have
$r\equiv 0\pmod {\mathfrak {p}}$
if and only if
$r\equiv 0\pmod {p}$
. We define a function
$m(p)$
on rational primes by
Consequently,
$\#\mathcal {N}^f(\mathfrak {p},X)=m(p)$
if
$c-a^2$
is a square in
$\mathcal {O}_K/\mathfrak {p}$
, and 0 otherwise.
The quadratic character of
$c-a^2$
in the residue field
$\mathcal {O}_K/\mathfrak {p}$
depends on the decomposition behaviour of the rational prime p in K. If p is inert, then
$\mathcal {O}_K/\mathfrak {p}=\mathbb {F}_{p^2}$
. Since every element of
$\mathbb {F}_p$
becomes a square in its quadratic extension
$\mathbb {F}_{p^2}$
, the integer
$c-a^2$
must be a square in
$\mathcal {O}_K/\mathfrak {p}$
. If p is splitting or ramified, then
$\mathcal {O}_K/\mathfrak {p}=\mathbb {F}_{p}$
. In this case, whether
$c-a^2$
is a square in
$\mathcal {O}_K/\mathfrak {p}$
depends on the Legendre symbol
$((c-a^2)/p)$
.
Using the Dedekind–Kummer theorem [Reference Neukirch12, Ch. 1, Proposition 8.3], the decomposition of a rational prime p in
$\mathcal {O}_K=\mathbb {Z}[\sqrt {-5}]$
is determined by the Legendre symbol
$({-5}/{p})$
. The prime p splits in
$\mathbb {Z}[\sqrt {-5}]$
if and only if
$({-5}/{p})=1$
, that is,
$p\equiv 1,3,7,9\pmod {20}$
. The prime p is inert in
$\mathbb {Z}[\sqrt {-5}]$
if and only if
$({-5}/{p})=-1$
, that is,
$p\equiv 11,13,17,19\pmod {20}$
. The discriminant of the field
$\mathbb {Q}[\sqrt {-5}]$
is
$d_K=-20$
. Thus, a prime p is ramified if and only if
$p\mid 20$
. Since we have assumed that p is coprime to
$2$
, the only possible ramified prime is
$p=5$
.
According to Theorem 1.2, the counting formula is given by
where
${\nu _{\mathfrak {n}}}$
is defined by
As illustrated in Example 2.5, the explicit formula for
$\#\mathcal {E}_{f,\mathfrak {n}}(X)$
relies heavily on the arithmetic properties of individual primes, such as their splitting behaviour in K and the quadratic character of the polynomial values. Consequently, the local intersection term
$\#\mathcal {N}^f(\mathfrak {p},X)$
varies irregularly, which makes the exact formula too complex for analysis of the global growth rate of the counting function. Motivated by this observation, we turn our attention in the next section to the asymptotic behaviour of
$\#\mathcal {E}_{f,\mathfrak {n}}(X)$
. By employing the Lang–Weil inequality, we can effectively control these local factors and establish a general bound that depends primarily on the geometric invariants of the variety X, rather than the specific arithmetic of each prime.
3 Asymptotic estimates for the counting function
In this section, we use the Lang–Weil inequality to give an approximation of
$\#\mathscr {E}_{f,\mathfrak {n}}(X)$
when X is geometrically irreducible over K and all primes dividing
$\mathfrak {n}$
are of good reduction for X. We first state the Lang–Weil inequality, which will be used repeatedly in this section.
Lemma 3.1 [Reference Lang and Weil7, Theorem 1], [Reference Ghorpade and Lachaud5, Remark 11.3]
Let
$k=\mathbb {F}_q$
be a finite field and X be a subvariety of
$\mathbb {A}^n_k$
with
$\dim X=r$
and
$\deg X=\ell $
. Then,
The implied constant in the O-term depends only on
$n,d$
and r.
Proof of Theorem 1.3
According to Theorem 1.2,
Recall that X is smooth and geometrically irreducible, and has a good reduction at
$\mathfrak {p}$
for any prime
$\mathfrak {p}\mid \mathfrak {n}$
. By the Lang–Weil inequality,
Let
$V(f)$
denote the set of zero points of the polynomial f. Since
$\mathcal {N}^f(\mathfrak {p},X)=X(\mathcal {O}_K/\mathfrak {p})\cap \bigcup _{i=1}^nV(f(x_i))$
, if there exists a prime ideal
$\mathfrak {p}\mid \mathfrak {n}$
such that
$X(\mathcal {O}_K/\mathfrak {p})\subset \mathcal {N}^f(\mathfrak {p},X)$
, then there exist no
$(f,\mathfrak {p})$
-exunits on
$X(\mathcal {O}_K/\mathfrak {p})$
. By Theorem 1.2, we have the trivial case
$\#\mathscr {E}_{f,\mathfrak {n}}(X)=0$
. If
$\mathcal {N}^f(\mathfrak {p},X)$
is a proper subset of
$X(\mathcal {O}_K/\mathfrak {p})$
, then each irreducible component of
$\mathcal {N}^f(\mathfrak {p},X)$
has dimension at most
$n-d-1$
. However, using the generalised Bézout theorem from intersection theory, the number of irreducible components of
$\mathcal {N}^f(\mathfrak {p},X)$
is uniformly bounded by a constant depending on
$d,n$
and the degree of f. Hence, using the Lang–Weil inequality,
The implied constant in the O-terms above depends only on
$K,X$
and f.
As a result,
We can bound the sum in the error term by
where
$\omega (\mathfrak {n})$
is the number of prime ideals of
$\mathcal {O}_K$
dividing
$\mathfrak {n}$
. Consequently, we obtain the estimate
This completes the proof of Theorem 1.3.
Lemma 3.1 implies that the coefficient of the dominant error term
$q^{r-{1}/{2}}$
vanishes whenever
$\deg (X) \le 2$
. This observation allows us to derive the sharper estimate presented in Theorem 1.4.
Proof of Theorem 1.4
We adopt the same notation and setup as in the proof of Theorem 1.3. Since
$\mathrm {deg}(X) \le 2$
, the coefficient
$(\mathrm {deg}X-1)(\mathrm {deg}X-2)$
in Lemma 3.1 is zero. Consequently, the estimate for the number of rational points on X improves to
For the intersection with the zero set of f, we retain the bound established in the previous proof, that is,
Combining these two estimates,
Since
$\sum _{\mathfrak {p}\vert \mathfrak {n}}\mathrm {N}_{K/\mathbb {Q}}(\mathfrak {p})^{-1}=O(\log \log (\mathrm {N}_{K/\mathbb {Q}}(\mathfrak {n})))$
,
This completes the proof of Theorem 1.4.
4 Concluding remarks
In this paper, we have established exact formulae and asymptotic estimates for the number of f-exunits on smooth algebraic varieties, assuming good reduction at all relevant primes.
A natural direction for future research is to extend these results to singular varieties or to analyse the distribution of solutions at primes of bad reduction. In such cases, the geometry of the fibres becomes significantly more complex, and the error terms in the Lang–Weil inequality may behave differently.
Furthermore, the similarity between number fields and function fields suggests that one can generalise these results to function fields.
Another interesting direction is to study the dynamical properties of the counting function. Harrington and Jones [Reference Harrington and Jones6] studied the iteration of
$\phi _e(n)$
in
$\mathbb {Z}_n$
, and Miguel [Reference Miguel9] suggested extending this to the ring of integers
$\mathcal {O}_K$
. However, defining a natural iteration for general f-exunits is complicated by the varying reduction types.
Acknowledgement
The authors are grateful to Zhengyu Tao for helpful discussions at the early stage of this work and for his valuable suggestions on the manuscript.
