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# 51% Attack via Difficulty Increase with a Small Quantum Miner

Bolton Bailey  Department of Computer Science, University of Illinois Urbana-
Champaign  Or Sattath  Department of Computer Science, Ben-Gurion University
of the Negev

###### Abstract

We present a strategy for a single quantum miner with relatively low hashing
power, with the same ramifications as a 51% attack. Bitcoin nodes consider the
chain with the highest cumulative proof-of-work to be the valid chain. A
quantum miner can manipulate the block timestamps to multiply the difficulty
by $c$. The fork-choice rule counts every block with increased difficulty with
weight $c$. By using Grover’s algorithm, it is only $O(\sqrt{c})$ harder for
the quantum miner to mine such blocks. By picking a high enough $c$, the
single quantum miner can create a competing chain with fewer blocks, but more
cumulative proof-of-work. The time required is $O(\frac{1}{r^{2}})$ epochs,
where $r$ is the fraction of the block rewards that the quantum miner would
have received if they mined honestly.

Most proof-of-work cryptocurrencies, including Bitcoin, are vulnerable to our
attack. However, it will likely be impossible to execute in forthcoming years,
as it requires an extremely fast and fault-tolerant quantum computer.

##  1 Introduction

It is often stated that a single quantum miner does not pose a security risk
to Bitcoin’s security. This position is supported by the formal analysis made
in [CGK+23], which shows some desired properties (namely, common prefix and
chain quality) are satisfied in a simplified model called the Bitcoin backbone
protocol in a network consisting of a single quantum miner and additional
(honest) classical miners.

This work shows that the contrary is true by exploiting an aspect that Ref.
[CGK+23] does not account for.

A single quantum miner, with relatively little hash power, can execute an
attack in the presence of honest classical miners with the same ramifications
as a 51% attack. Specifically, the attacker can double-spend transactions and
censor others, thus breaking Bitcoin’s security, and can receive $1-\epsilon$
of the block rewards for any $\epsilon>0$.

The attack relies on the “chain work” consensus mechanism in Bitcoin and many
other proof-of-work (PoW) based cryptocurrencies: In case of competing forks,
Bitcoin nodes follow the branch which has the most accumulated PoW, see
[Nak08, Section 4] and [Ant17, pp. 239–240]. In short forks, where different
branches have the same difficulty, this coincides with the longest chain rule
but may not be so in long forks, which span over more than one epoch and with
different difficulty adjustments: the sum of difficulties in the shorter chain
may be higher than that in the longer chain. This consensus mechanism thwarts
certain classical attacks (we discuss one such attack in more detail in
Appendix B), but it leaves the protocol vulnerable to quantum attacks.

######  Remark 1.

Our results do not contradict those in [CGK+23]: as mentioned, their results
hold in a _simplified_ model. In their model, there are no difficulty
adjustments; therefore, the longest chain rule and the chain with the most
cumulative PoW coincide. In practice, a difficulty adjustment mechanism must
be introduced; otherwise, due to fluctuations in the miners’ hash power, the
rate at which blocks are created can be either higher or slower than desired.
An earlier version of their work lists difficulty adjustments as an important
future direction: “The first generalization to consider is the Bitcoin
backbone protocol with variable difficulty” [CGK+19, Section 6]. As they point
out, such a generalization _was_ considered in the classical setting [GKL17];
our work shows that the quantum generalization does not apply.

######  Remark 2.

This work studies a setting with a _single_ quantum miner. The fully quantum
setting, where all miners use quantum hardware, is mostly uncharted territory.
Ref. [Sat20] shows that Bitcoin’s tie-breaking rule needs to be adapted. Ref.
[LRS19] analyzes the equilibria strategies for quantum miners in a simplified
model.

The main advantage quantum miners have over classical miners is that they can
use Grover’s algorithm. We now explain the essential properties of Grover’s
algorithm for this work at a high level.

Let $f:\\{1,\ldots,n\\}\to\\{0,1\\}$. We call $x$ a marked item if and only if
$f(x)=1$, and denote by $k$ is the number of the marked items in $f$. We
assume that the function $f$ is modeled as a black box, and every query made
to the function is counted. Classically, it can easily be shown that one needs
$\theta(\frac{n}{k})$ queries to the function to find a marked item with
constant probability. Grover’s algorithm gives a quadratic speed-up. That is,
the quantum algorithm finds a marked item with high probability using only
$\Theta(\sqrt{\frac{n}{k}})$ quantum queries to the function. Throughout this
work, for the sake of simplicity, we ignore the constant factor in the
$\Theta(\cdot)$ notation and the probabilistic nature of Grover’s algorithm
and treat it as if its running time is $\sqrt{\frac{n}{k}}$ and success
probability is $1$. In the case of mining, the function $f$ is defined as the
function that returns $1$ if the hash of a block with a nonce $x$ is below the
difficulty, and furthermore, we model the hash function as in the random
oracle model. No further understanding of quantum computing or Grover’s
algorithm will be needed. The interested reader is referred to Ref. [NC00] for
an in-depth introduction to quantum computing.

The most important insight for the attack is the following: A difficulty
increase by a factor of $c$ makes it $c$ times harder for classical miners to
find a block, but only $\sqrt{c}$ harder for quantum miners using Grover’s
algorithm to do so. A block with such increased difficulty is accounted as $c$
blocks with the original difficulty for the accumulated PoW, but is only
$\sqrt{c}$ harder for for the quantum miner to find. By increasing the
difficulty by a factor of $c$, a quantum miner can increase the rate at which
they add cumulative work to a chain by a factor of $\sqrt{c}$.

We will describe the main difficulty increase attack in this section and three
more variants in Section 2, each improving or addressing a problem in the
previous variant. The only downside of these improved variants is that they
take longer to execute.

For concreteness, we use a running example with the same parameters as in
Bitcoin: the expected time to create a block by the classical miners is 10
minutes, and difficulty adjustments are made every epoch, which lasts 2016
blocks. Creating an epoch worth of blocks takes two weeks if a block is
created every 10 minutes, and we refer to this period as _epoch-time_.111 Note
that it may take much longer than one epoch-time for the quantum miner to mine
one epoch of blocks, especially as the difficulty in that epoch increases. We
further assume that the quantum miner can mine a block every $40$ minutes. In
other words, by running Grover’s algorithm, the miner can create a block at
$r=\frac{1}{4}$ of the speed relative to the classical miners.222In our
running example, we label numbers derived from this choice of $r$ in blue.

######  Remark 3.

Note that a quantum miner that can generate a block at a ratio $r$ compared to
the classical miners, might generate less than $r$ of the total revenue from
these blocks, due to stale blocks. Stale blocks are blocks that the quantum
miner generates but would eventually not be part of the longest chain. This
phenomenon has some surprising consequences observed and analyzed by Nerem and
Gaur [NG23], but will not be relevant to our work.

#### Main attack.

The attacker’s first task is to increase the difficulty by a factor of
$\frac{4}{r^{2}}={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}64}$,
where $r=\frac{1}{4}$ mentioned above. To achieve this first task, the
attacker starts to mine an entire epoch with fake timestamps such that the
blocks seem to be created
$\frac{4}{r^{2}}={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}64}$
times more often than they should be. Recall that difficulty adjustments aim
to ensure blocks are created at a designated rate: one block is expected every
$10$ minutes in Bitcoin. At the end of the epoch, since the timestamps in
these blocks appeared to occur every
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\frac{10}{64}}$
minutes, the difficulty in the attacker’s chain is increased by a factor of
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}64}=\frac{4}{r^{2}}$,
as desired.333This is not entirely accurate, see p. 2.3, and Variant 4 which
resolves this issue.

The miner’s second task is to create another epoch of blocks at the new
difficulty. In more detail, the miner creates one epoch worth of blocks—that
is 2016 blocks—with the new difficulty (which is $64$ times harder than the
original one), by spacing the timestamps at the usual 10 minutes. The miner
publishes these blocks, while ignoring blocks created by the classical miners.

We claim that after the first epoch with the increased difficulty is
published, the quantum miner has a chain with more accumulated PoW than the
classical miners. To see this, let us first calculate how long it takes for
the quantum miner to achieve the first goal. Recall that it takes
$\frac{10}{r}={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}40}$
minutes to mine a single block with the original difficulty by the quantum
miner, which is $4$ times as much the classical miners take in expectation.
So, it will take
$\frac{1}{r}={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}4}$
epoch-times (8 weeks) to achieve the first goal by the quantum miner.

The difficulty of the second epoch is
$\frac{4}{r^{2}}={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}64}$
times higher, so mining it would take the attacker
$\sqrt{\frac{4}{r^{2}}}=\frac{2}{r}={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}8}$
times more than the first epoch; in other words, quantum mining an epoch with
the new difficulty will take
$\frac{2}{r}\cdot\frac{1}{r}=\frac{2}{r^{2}}={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}32}$
epoch-times.

The quantum miner accumulated
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}65}=1+\frac{4}{r^{2}}\geq\frac{4}{r^{2}}$
epochs worth of PoW measured in the original difficulty. The time it took is
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}36}=\frac{1}{r}+\frac{2}{r^{2}}\leq\frac{3}{r^{2}}$
epochs. So, we see that it took the quantum miner at most $\frac{3}{r^{2}}$
epoch-times, to generate at least $\frac{4}{r^{2}}$ epochs worth of PoW.
Therefore, the quantum miner’s chain has at least $\frac{1}{r^{2}}$ more
cumulative PoW than the expected cumulative PoW by the classical miners, so
with high probability, the quantum miner’s chain will be the chain with the
most accumulated PoW.

This variant allows the quantum miner to double spend a transaction: the
quantum miner can mine his chain privately and, while mining, spend a
transaction on the classical miners’ chain, and post a conflicting transaction
on his chain. After this transaction is confirmed (since the quantum miner is
mining in private, the transaction will have many confirmations), and the
goods are received, the miner can publish his private chain, which will
eventually have more PoW, hence causing a long-term reorganization, causing
the original transfer to get canceled. The details of the Variant 1 attack are
depicted in Table 1.

n | difficulty | CPoW | timeToCreate | realTimeWhenCreated | timestamp
---|---|---|---|---|---
$1$ | $1$ | $1$ | $4$ | $4$ | $1.56\cdot 10^{-2}$
$2$ | $64$ | $65$ | $32$ | $36$ | $1.02$

Table 1: The main attack. The column headers represent the following: n is the
epoch number, difficulty is the difficulty of that epoch relative to the
original difficulty, CPoW is the cumulative Proof-of-Work of the quantum
miner, timeToCreate is the time the quantum miner invested in creating this
epoch, measured in epoch-time (for example, 1 represents 2 weeks for Bitcoin),
realTimeWhenCreated is the real time, measured in epoch-times, when that epoch
was created, and timestamp represents the timestamps that appear in the last
block in that epoch of the quantum miner, measured in epoch-time. All the
other tables have the same format.

The quantum miner can keep mining at the new difficulty, while ignoring the
blocks generated by the classical miners, and produce $100\%$ of the blocks in
the chain with the most cumulative PoW.

This strategy has several issues that are discussed and addressed in several
variants in Section 2. The first issue is that the time-stamps are lagging:
the timestamps in the blocks generated by the quantum miner lag behind real
time; This issue is addressed in Variant 2. The next issue is related to the
incentives. The quantum miner that uses the strategy above creates only 2
epochs worth of blocks, even though the total time of the attack is 36 epochs.
In Variant 3, we show an attack where the quantum miner’s revenue gets
arbitrarily close to the maximal revenue, namely, the revenue generated by an
honest classical miner with $100\%$ of the hashing power. In the variants
above, we apply sharp changes to the mining difficulty. Bitcoin limits the
increase and decrease factor of the difficulty by a factor of four. This can
be easily addressed, by making smaller changes to the difficulty, as
demonstrated in Variant 4. The main drawback of these variants compared to the
attack above, is that they take longer to perform, though, asymptotically,
they all take $O\left(\frac{1}{r^{2}}\right)$ epoch-times to implement.

##  2 Variants

###  2.1 The Lagging Issue

Even though the quantum miner can create a chain with higher cumulative PoW,
the timestamps in the chain in the variant above are lagging behind the real
time. More specifically, in the example above, one can notice that the
timestamp in the second row is 1.02 epoch-times, whereas the real time is 36
epoch-times. Therefore, the attack can easily be noticed. Furthermore, an
attempt to rule out the attack can be made, by changing the fork choice rule:
blocks that have timestamps that substantially lag behind the real time are
ignored by honest nodes.

In the 2nd variant, when the quantum attacker’s chain is published, the final
time-stamp does not lag behind the real time. The modified fork-choice rule
would permit the fork of the quantum miner and, therefore, is ineffective for
the second variant.

#### Variant 2.

The goal of this variant is to resolve the lag that is created in the first
variant. Table 2 shows the details of the attack. The attack proceeds in three
phases: After the difficulty is increased in the first phase, as in Variant 1,
there is a second phase in which the quantum miner _reduces_ the difficulty,
to a lower level than the original difficulty, which allows the quantum miner
to generate blocks much faster in the third phase. Specifically, the
difficulty is ultimately reduced to
$r^{6}={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\frac{1}{4^{6}}}$.444While
it is technically only necessary to reduce the difficulty to below $r^{2}$ to
make up ground on the timestamp gap, our choice of $r^{6}$ makes the analysis
more clean. This is done by reducing the current difficulty (which is
$\frac{4}{r^{2}})$ by a factor of 8 for
$\log_{8}(\frac{4}{r^{8}})={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}6}$
epochs. 555Note that the alternative in which the difficulty is decreased in 1
epoch by this factor would take much longer: It would require making an
average timestamp increase of $\frac{4}{r^{8}}$ block times, leading to a
timestamp in the future. This solves the issue at hand, where the timestamps
in the attacker’s blocks are noticeably behind the real time, but introduces
the _opposite_ problem. The way to reduce the difficulty by a factor $d$ (up
to rounding errors) while inducing the least change in timestamp is to reduce
the difficulty for $\ln(d)$ epochs, where the difficulty decreases by a factor
$e$ between every two epochs.

The time that it takes to mine the hardest epoch is
$\frac{2}{r^{2}}={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}32}$
epoch-times, and every consecutive epoch has an easier difficulty by a factor
8, so each consecutive block takes only a fraction of $\frac{1}{\sqrt{8}}$ the
time compared to the previous epoch. Overall, it takes

$\frac{2}{r^{2}}\sum_{i=0}^{\log_{8}(\frac{4}{r^{8}})-1}\left(\frac{1}{\sqrt{8}}\right)^{i}\leq\frac{2}{r^{2}}\sum_{i=0}^{\infty}\left(\frac{1}{\sqrt{8}}\right)^{i}=\frac{2}{r^{2}(1-\frac{1}{\sqrt{8}})}$

epoch-times for the quantum miner to reduce the difficulty to the easy level.

Recall that the quantum miner can generate each block at the original
difficulty every
$\frac{1}{r}={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}4}$
blocks. Due to the decreased difficulty, and Grover’s algorithm running time,
mining a block at that reduced difficulty takes only
$\frac{\sqrt{r^{6}}}{r}=r^{2}={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\frac{1}{16}}$
epoch-times. The timestamps are kept at the normal pace (i.e., a block every
10 minutes), so the lag is getting shorter. The quantum miner keeps mining
until there is no significant gap between the timestamp and when the block is
created. In this example, as seen in Table 2, the lag is shortened by mining
$5$ additional blocks at the easy difficulty level. As can be seen, there is
no significant lag between the timestamp and the time the blocks are
published: 53.72 vs 53.02 epoch-times (and can be made arbitrarily small
easily).

Recall that the goal of the last phase is to make sure that there is no lag
between the timestamp and the real-time in the last epoch of the attack. The
time it takes for the quantum miner to mine each epoch at the easy difficulty
is $\sqrt{r^{6}}\cdot\frac{1}{r}=r^{2}$ epoch-times, whereas the timestamp
grows by exactly 1 epoch-time each epoch, so the gap between the timestamp and
the real time shrinks by $1-r^{2}$ epoch-times each mined epoch. The total
time that needs to be gapped is at most the total time of the attack in the
first two phases, which is upper-bounded by
$(\frac{1}{r}+\frac{2}{r^{2}(1-\frac{1}{\sqrt{8}})})$. So, the total time
devoted by the quantum miner to mine at the easy difficulty is at most
$(\frac{1}{r}+\frac{2}{r^{2}(1-\frac{1}{\sqrt{8}})})\cdot\frac{r^{2}}{1-r^{2}}$
epoch-times. The overall epoch-time of the attack (that is, by adding the time
for the previous phase) is

$\left(\frac{1}{r}+\frac{2}{r^{2}(1-\frac{1}{\sqrt{8}})}\right)\cdot\left(1+\frac{r^{2}}{1-r^{2}}\right)\leq\frac{3.57}{r^{2}},$

where we used $r\leq\frac{1}{4}$ in the last inequality.

######  Remark 4.

Notice that for attackers with a value of $r$ close to 1, the attack as laid
out takes a long time, because we do not reduce the difficulty to a very small
value. For this reason it is convenient to assume that an attacker with
$r>\frac{1}{4}$ simply carries out the attack as if $r=\frac{1}{4}$, so that
they decrease the difficulty sufficiently in the latter stage. This preserves
the $O(\frac{1}{r^{2}})$ asymptotics.

So, the expected cumulative PoW that the honest network can generate is at
most as above. The quantum miner generates $\frac{4}{r^{2}}$ PoW in the second
epoch of the attack alone, so the accumulated PoW on the quantum attacker’s
chain is greater than the expected accumulated PoW of the classical miners.

n | difficulty | CPoW | timeToCreate | realTimeWhenCreated | timestamp
---|---|---|---|---|---
$1$ | $1$ | $1$ | $4$ | $4$ | $1.56\cdot 10^{-2}$
$2$ | $64$ | $65$ | $32$ | $36$ | $8.02$
$3$ | $8$ | $73$ | $11.31$ | $47.31$ | $16.02$
$4$ | $1$ | $74$ | $4$ | $51.31$ | $24.02$
$5$ | $0.13$ | $74.13$ | $1.41$ | $52.73$ | $32.02$
$6$ | $1.56\cdot 10^{-2}$ | $74.14$ | $0.5$ | $53.23$ | $40.02$
$7$ | $1.95\cdot 10^{-3}$ | $74.14$ | $0.18$ | $53.4$ | $48.02$
$8$ | $2.44\cdot 10^{-4}$ | $74.14$ | $6.25\cdot 10^{-2}$ | $53.47$ | $49.02$
$9$ | $2.44\cdot 10^{-4}$ | $74.14$ | $6.25\cdot 10^{-2}$ | $53.53$ | $50.02$
$10$ | $2.44\cdot 10^{-4}$ | $74.14$ | $6.25\cdot 10^{-2}$ | $53.59$ | $51.02$
$11$ | $2.44\cdot 10^{-4}$ | $74.14$ | $6.25\cdot 10^{-2}$ | $53.65$ | $52.02$
$12$ | $2.44\cdot 10^{-4}$ | $74.14$ | $6.25\cdot 10^{-2}$ | $53.72$ | $53.02$

Table 2: The table shows the details of Variant 2. Note that in the strategy,
the timestamp in epoch 12 is $53.02$ epoch-times, while the real time lags
only slightly behind it $53.72$. This should be compared vis-a-vis the first
variant, where the lag was substantially higher. Note that during that 53
epoch-times, the miner has mined only 12 epochs worth of blocks. Therefore,
the block subsidy from the quantum mining relative to a miner that mines all
the blocks with the original difficulty is only $23\%$; this will be improved
in the third variant.

###  2.2 The Revenue Issue

In Variant 2, the attacker creates only
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}12}$ epochs
worth of blocks, even though in real time, the attack takes about
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}53}$ epoch-
times. This means that the revenue during the attack from the block subsidy is
only
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\approx
23\%}$ of the total revenue that can be generated from mining at the original
speed. Typically, a 51% attacker would accumulate 100% of the expected block
subsidy for the time period they carry out the attack.

Next, we show a variant in which the revenue from the block subsidy can get
arbitrarily close to the total revenue that can be extracted by a classical
miner with 100% of the hashing power.

#### Variant 3.

Increasing the block subsidy revenue is done by adding several epochs in the
increased difficulty, and additional epochs in the decreased difficulty, until
the timestamp and the time in which the quantum miner’s chain is published are
close enough. In the example shown in Table 3, three epochs are mined at the
increased difficulty (compared to one epoch in Variant 2), and 72 epochs with
the decreased difficulty (compared to five epochs in Variant 2). The total
time in this variant is 121.84 epochs. During it, the quantum miner generates
80 epochs worth of blocks, yielding approximately
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}66\%}$ of
the block subsidy revenue that the entire network, mining at the original
speed, would have generated. This improves upon the $23\%$ of the block
subsidy revenue generated in Variant 2. In Appendix A we discuss this attack
in more generality, where the attack achieves $1-\epsilon$ fraction of the
revenue generated honestly, for an arbitrary small $\epsilon$, and takes
$O(\frac{1}{\epsilon r^{2}})$ epochs.

n | difficulty | CPoW | timeToCreate | realTimeWhenCreated | timestamp
---|---|---|---|---|---
$1$ | $1$ | $1$ | $4$ | $4$ | $1.56\cdot 10^{-2}$
$2$ | $64$ | $65$ | $32$ | $36$ | $1.02$
$3$ | $64$ | $129$ | $32$ | $68$ | $2.02$
$4$ | $64$ | $193$ | $32$ | $100$ | $10.02$
$5$ | $8$ | $201$ | $11.31$ | $111.31$ | $18.02$
$6$ | $1$ | $202$ | $4$ | $115.31$ | $26.02$
$7$ | $0.13$ | $202.13$ | $1.41$ | $116.73$ | $34.02$
$8$ | $1.56\cdot 10^{-2}$ | $202.14$ | $0.5$ | $117.23$ | $42.02$
$9$ | $1.95\cdot 10^{-3}$ | $202.14$ | $0.18$ | $117.4$ | $50.02$
$10$ | $2.44\cdot 10^{-4}$ | $202.14$ | $6.25\cdot 10^{-2}$ | $117.47$ | $51.02$
$11$ | $2.44\cdot 10^{-4}$ | $202.14$ | $6.25\cdot 10^{-2}$ | $117.53$ | $52.02$
$12$ | $2.44\cdot 10^{-4}$ | $202.14$ | $6.25\cdot 10^{-2}$ | $117.59$ | $53.02$
$13$ | $2.44\cdot 10^{-4}$ | $202.14$ | $6.25\cdot 10^{-2}$ | $117.65$ | $54.02$
$14$ | $2.44\cdot 10^{-4}$ | $202.14$ | $6.25\cdot 10^{-2}$ | $117.72$ | $55.02$
|  |  | ⋮ |  |
$78$ | $2.44\cdot 10^{-4}$ | $202.16$ | $6.25\cdot 10^{-2}$ | $121.72$ | $119.02$
$79$ | $2.44\cdot 10^{-4}$ | $202.16$ | $6.25\cdot 10^{-2}$ | $121.78$ | $120.02$
$80$ | $2.44\cdot 10^{-4}$ | $202.16$ | $6.25\cdot 10^{-2}$ | $121.84$ | $121.02$

Table 3: The table shows the details of Variant 3. Note that epochs 15–77 were
omitted; their difficulty is the same as epoch 14, and the effect is similar
to the epochs below and above them.

###  2.3 The Incompatible Difficulty Adjustments Issue

In some cryptocurrencies the shifts in the PoW difficulty adjustments are
bounded. For example, in Bitcoin, the difficulty between two consecutive
epochs cannot666See
https://github.com/bitcoin/bitcoin/blob/4b1196a9855dcd188a24f393aa2fa21e2d61f061/src/pow.cpp#L56.
increase by more than a factor of $4$, or less than $\frac{1}{4}$. In all the
previous variants, we increased the difficulty dramatically in the first epoch
and reduced it by a factor of 8 in several consecutive epochs: all these
transitions are incompatible with Bitcoin’s difficulty adjustments. This
aspect is addressed in the following variant.

n | difficulty | CPoW | timeToCreate | realTimeWhenCreated | timestamp
---|---|---|---|---|---
$1$ | $1$ | $1$ | $4$ | $4$ | $0.5$
$2$ | $2$ | $3$ | $5.66$ | $9.66$ | $1$
$3$ | $4$ | $7$ | $8$ | $17.66$ | $1.5$
$4$ | $8$ | $15$ | $11.31$ | $28.97$ | $2$
$5$ | $16$ | $31$ | $16$ | $44.97$ | $2.5$
$6$ | $32$ | $63$ | $22.63$ | $67.6$ | $3$
$7$ | $64$ | $127$ | $32$ | $99.6$ | $3.5$
$8$ | $128$ | $255$ | $45.25$ | $144.85$ | $4$
$9$ | $256$ | $511$ | $64$ | $208.85$ | $5$
$10$ | $256$ | $767$ | $64$ | $272.85$ | $6$
$11$ | $256$ | $1{,}023$ | $64$ | $336.85$ | $8$
$12$ | $128$ | $1{,}151$ | $45.25$ | $382.11$ | $10$
$13$ | $64$ | $1{,}215$ | $32$ | $414.11$ | $12$
$14$ | $32$ | $1{,}247$ | $22.63$ | $436.74$ | $14$
$15$ | $16$ | $1{,}263$ | $16$ | $452.74$ | $16$
$16$ | $8$ | $1{,}271$ | $11.31$ | $464.05$ | $18$
$17$ | $4$ | $1{,}275$ | $8$ | $472.05$ | $20$
$18$ | $2$ | $1{,}277$ | $5.66$ | $477.71$ | $22$
$19$ | $1$ | $1{,}278$ | $4$ | $481.71$ | $24$
$20$ | $0.5$ | $1{,}278.5$ | $2.83$ | $484.53$ | $26$
$21$ | $0.25$ | $1{,}278.75$ | $2$ | $486.53$ | $28$
$22$ | $0.13$ | $1{,}278.88$ | $1.41$ | $487.95$ | $30$
$23$ | $6.25\cdot 10^{-2}$ | $1{,}278.94$ | $1$ | $488.95$ | $32$
$24$ | $3.13\cdot 10^{-2}$ | $1{,}278.97$ | $0.71$ | $489.66$ | $34$
$25$ | $1.56\cdot 10^{-2}$ | $1{,}278.98$ | $0.5$ | $490.16$ | $36$
$26$ | $7.81\cdot 10^{-3}$ | $1{,}278.99$ | $0.35$ | $490.51$ | $38$
$27$ | $3.91\cdot 10^{-3}$ | $1{,}279$ | $0.25$ | $490.76$ | $40$
$28$ | $1.95\cdot 10^{-3}$ | $1{,}279$ | $0.18$ | $490.94$ | $42$
$29$ | $9.77\cdot 10^{-4}$ | $1{,}279$ | $0.13$ | $491.06$ | $43$
$30$ | $9.77\cdot 10^{-4}$ | $1{,}279$ | $0.13$ | $491.19$ | $44$
$31$ | $9.77\cdot 10^{-4}$ | $1{,}279$ | $0.13$ | $491.31$ | $45$
|  |  | ⋮ |  |
$539$ | $9.77\cdot 10^{-4}$ | $1{,}279.5$ | $0.13$ | $554.81$ | $553$
$540$ | $9.77\cdot 10^{-4}$ | $1{,}279.5$ | $0.13$ | $554.94$ | $554$
$541$ | $9.77\cdot 10^{-4}$ | $1{,}279.5$ | $0.13$ | $555.06$ | $555$

Table 4: The table shows the details of Variant 4. As can be seen, the
difficulty adjustments in this variant are at most $2$ and at least
$\frac{1}{2}$, compared to the previous variants, where the increases in
difficulty were much higher, and the decrease in difficulty reached a factor
of $\frac{1}{8}$; these sharp changes are not in the valid regime of Bitcoin’s
difficulty adjustments, which is $[\frac{1}{4},4]$. Note that the relative
revenue in this variant increased to $\frac{541}{555.06}\approx 97\%$.

#### Variant 4.

As mentioned above, we need to reduce the rate of change in the difficulty
adjustments: Instead of increasing the difficulty in the first epoch, we do so
by increasing the difficulty in several epochs, where the difficulty is
increased each epoch by a factor of 2; Similarly, instead of reducing the
difficulty by a factor of 8, we reduce it by a factor of 2, which triples the
number of epochs needed for that difficulty reduction.

The attack is shown in Table 4. Increasing by a factor of 2 in each epoch, it
takes 8 epochs to reach the maximum difficulty of 256, and decreasing by a
factor of 2 each epoch, it then takes 18 epochs to reach the minimum
difficulty of $\approx 9.77\cdot 10^{-4}$. Ultimately, though, these
transitions only require a number of epochs logarithmic in the desired
difficulty change, and their effect on the cumulative PoW and timestamp change
is dominated by the contributions from the blocks at the top and bottom
difficulties, leaving the total running time still at $O(\frac{1}{r^{2}})$.

##  3 Applicability

As far as we are aware, in theory, the attack applies to all existing PoW
cryptocurrencies such as Bitcoin [Nak08], as well as variants thereof, which
all need to have some form of difficulty adjustments [SZ15, PS17].

The attack described in this work cannot be executed with existing
technologies because of two main reasons: (i) The quantum computers have on
the order of (only) 100 qubits, and (ii) these quantum computers are too noisy
to perform long computations.

Next, we analyze the minimal requirements of a quantum computer to apply this
attack. As of February 2024, Bitcoin hash-rate is around 500 exa-hashes per
second (EH/S)$=5\cdot 10^{20}$ H/S.777See, e.g.,
https://www.coinwarz.com/mining/bitcoin/hashrate-chart Since a block is
created every 600 seconds, it means that the probability to win a block per
hash is $1/(600\cdot 5\cdot 10^{20})=\frac{1}{3\cdot 10^{23}}$. To find a
block, a miner needs to apply the order of $\sqrt{3\cdot 10^{23}}$ Grover
iterations. Implementing a SHA-256 evaluation can be done via a circuit with a
depth of 1600, see https://nigelsmart.github.io/MPC-Circuits/. So, for a miner
to mine a block on a single device, requires a quantum computation with depth
in the order of $1600\cdot\sqrt{3\cdot 10^{23}}\lesssim 10^{15}$. Suppose a
fast, noiseless quantum computer operates at a 10GHz clock speed. Such a
quantum computer would find a block every $10^{15}/10^{10}=10^{5}$ seconds,
which is roughly once a day. This is about $140$ times slower than the
existing block rate. So, naively, such a single quantum computer would have
$r\approx 0.7\%$ of the hash-rate of the network. Recall that the attack
proposed in our work takes the order of $r^{-2}$ epochs $\approx 800$ years,
which is clearly unrealistic under the assumptions above. There are several
routes in which the attack could become realistic: (i) the clock speed of the
quantum computer has to be more than an order of magnitude faster than the
fastest commercially available classical computers. (ii) Attack
cryptocurrencies with less total hash-rate, or much shorter epoch-time. (iii)
Use many quantum computers in parallel in the attack. Note that Grover’s
algorithm does not lend itself to perfect parallelization as classical
unordered search: Running Grover’s algorithm on $k$ quantum computers cuts the
time only by a $\frac{1}{\sqrt{k}}$ multiplicative factor [Zal99, Section IV].

##  4 Conclusion and Open problems

Our work has demonstrated a variety of attacks on Nakamoto consensus using
difficulty adjustment. These attacks are not possible or practical with the
quantum computers of today. In the long-term though, they represent a
challenge to the thesis that 51% of network hashpower is necessary to execute
a double-spend on Bitcoin. At the least, our work shows that hopes of a
security proof for this consensus system that realizes both quantum and
difficulty adjustment will not be realized.

We conclude with some potential directions for future work:

#### Impossibility results for more general PoW consensus mechanisms

Our work here deals with standard Nakamoto consensus. However, there are a
variety of other forms of consensus that use PoW which involve multiple chains
or even directed-acyclic graph structures [BKT+19, PS17, SZ15, SLZ16]. While
the cited works do not always cover difficulty adjustment, it seems likely
that it could be added in some form to most of these other consensus
mechanisms. Future work could explore whether there are stumbling blocks akin
to the issues covered in Section 2 that prevent the difficulty increase attack
for these mechanisms. In particular, do limitations on the timestamp deltas or
bounds on per-epoch difficulty change help prevent the attack in these
systems?

#### Optimal attacks under different attacker preferences

Malicious actors may want to attack blockchains for a variety of reasons. At a
large scale, nation-states may be opposed to blockchains for reasons relating
to financial regulation and control. On a smaller scale, there may be criminal
organizations who simply want to extract monetary value from the attack,
either by double-spending or by using the event of a chain compromise to short
the market. The plethora of attack models imply different circumstances as to
the resources of the attackers, and their relative preference on the speed and
value of the attack. Future work could consider these possibilities in more
specificity, and determine the optimal attack parameters for specific goals.

#### Mitigations

As we discussed in Section 3, one of the limiting factors on the attack is the
size of the circuit used for the hash function. This suggests one way of
mitigating an attack of the type we propose - simply making the hash function
more difficult to compute. This is unfortunately only a stopgap, though, since
Grover’s algorithm can ultimately be applied to any PoW system that takes the
form of a repeated search problem. Future work might consider if there is some
way of structuring the problems that miners solve so that they cannot be sped
up by Grover search. Alternatively, one could consider if there are other ways
of structuring the difficulty adjustment and the relative weights assigned to
blocks of different difficulty that defeat this type of attack.

### Acknowledgments

The authors would like to thank Andrew Miller for discussions on the topic of
this paper.

This work was funded by the European Union (ERC-2022-COG, ACQUA, 101087742).
Views and opinions expressed are however those of the author(s) only and do
not necessarily reflect those of the European Union or the European Research
Council Executive Agency. Neither the European Union nor the granting
authority can be held responsible for them.

This material is based upon work supported by the National Science Foundation
under the Graduate Research Fellowship Program with Grant No. DGE – 1746047.

## References

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  * [BKT+19] V. Bagaria, S. Kannan, D. Tse, G. Fanti, and P. Viswanath.  Prism: Deconstructing the blockchain to approach physical limits.  In Proceedings of the 2019 ACM SIGSAC Conference on Computer and Communications Security, pages 585–602, 2019.
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  * [LRS19] T. Lee, M. Ray, and M. Santha.  Strategies for Quantum Races.  In A. Blum, editor, 10th Innovations in Theoretical Computer Science Conference, ITCS 2019, volume 124 of LIPIcs, pages 51:1–51:21. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2019, arXiv: 1809.03671.
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##  Appendix A Increased revenue

Recall that Variant 3 allows a quantum miner to increase the revenue up to
$(1-\epsilon)b$, where $b$ is the revenue that an honest miner controlling
$100\%$ of the hashing power will generate during that period. Table 5
demonstrates an attack with $\epsilon=0.05$, i.e., the quantum miner generates
$95\%$ of the total revenue. A similar analysis to that in Variant 2 shows
that the time of the attack scales as $O(\frac{1}{\epsilon r^{2}})$ epochs.

n | difficulty | CPoW | timeToCreate | realTimeWhenCreated | timestamp
---|---|---|---|---|---
$1$ | $1$ | $1$ | $4$ | $4$ | $1.56\cdot 10^{-2}$
$2$ | $64$ | $65$ | $32$ | $36$ | $1.02$
$3$ | $64$ | $129$ | $32$ | $68$ | $2.02$
$4$ | $64$ | $193$ | $32$ | $100$ | $3.02$
$5$ | $64$ | $257$ | $32$ | $132$ | $4.02$
$6$ | $64$ | $321$ | $32$ | $164$ | $5.02$
$7$ | $64$ | $385$ | $32$ | $196$ | $6.02$
$8$ | $64$ | $449$ | $32$ | $228$ | $7.02$
$9$ | $64$ | $513$ | $32$ | $260$ | $8.02$
$10$ | $64$ | $577$ | $32$ | $292$ | $9.02$
$11$ | $64$ | $641$ | $32$ | $324$ | $10.02$
$12$ | $64$ | $705$ | $32$ | $356$ | $11.02$
$13$ | $64$ | $769$ | $32$ | $388$ | $12.02$
$14$ | $64$ | $833$ | $32$ | $420$ | $13.02$
$15$ | $64$ | $897$ | $32$ | $452$ | $14.02$
$16$ | $64$ | $961$ | $32$ | $484$ | $15.02$
$17$ | $64$ | $1{,}025$ | $32$ | $516$ | $16.02$
$18$ | $64$ | $1{,}089$ | $32$ | $548$ | $17.02$
$19$ | $64$ | $1{,}153$ | $32$ | $580$ | $18.02$
$20$ | $64$ | $1{,}217$ | $32$ | $612$ | $19.02$
$21$ | $64$ | $1{,}281$ | $32$ | $644$ | $20.02$
$22$ | $64$ | $1{,}345$ | $32$ | $676$ | $21.02$
$23$ | $64$ | $1{,}409$ | $32$ | $708$ | $22.02$
$24$ | $64$ | $1{,}473$ | $32$ | $740$ | $23.02$
$25$ | $64$ | $1{,}537$ | $32$ | $772$ | $31.02$
$26$ | $8$ | $1{,}545$ | $11.31$ | $783.31$ | $39.02$
$27$ | $1$ | $1{,}546$ | $4$ | $787.31$ | $47.02$
$28$ | $0.13$ | $1{,}546.13$ | $1.41$ | $788.73$ | $55.02$
$29$ | $1.56\cdot 10^{-2}$ | $1{,}546.14$ | $0.5$ | $789.23$ | $63.02$
$30$ | $1.95\cdot 10^{-3}$ | $1{,}546.14$ | $0.18$ | $789.4$ | $71.02$
$31$ | $2.44\cdot 10^{-4}$ | $1{,}546.14$ | $6.25\cdot 10^{-2}$ | $789.47$ | $72.02$
$32$ | $2.44\cdot 10^{-4}$ | $1{,}546.14$ | $6.25\cdot 10^{-2}$ | $789.53$ | $73.02$
|  |  | ⋮ |  |
$33$ | $2.44\cdot 10^{-4}$ | $1{,}546.14$ | $6.25\cdot 10^{-2}$ | $789.59$ | $74.02$
$794$ | $2.44\cdot 10^{-4}$ | $1{,}546.33$ | $6.25\cdot 10^{-2}$ | $837.15$ | $835.02$
$795$ | $2.44\cdot 10^{-4}$ | $1{,}546.33$ | $6.25\cdot 10^{-2}$ | $837.22$ | $836.02$
$796$ | $2.44\cdot 10^{-4}$ | $1{,}546.33$ | $6.25\cdot 10^{-2}$ | $837.28$ | $837.02$

Table 5: The table shows another variant, where the relative revenue is
$95\%$, which improves upon the $66\%$ achieved in Variant 3. This variant is
closely related to Variant 3, except it has 24 epochs mined at the highest
difficulty (compared to only 3 epochs in Variant 3), and additional blocks at
the easiest difficulty to reduce the lag.

##  Appendix B Problems with the longest chain rule

Our attack crucially relies on Bitcoin’s fork-choice rule, which prefers the
chain with the most cumulative PoW. One may consider a natural alternative, in
which the fork-choice rule is changed to the longest-chain, ignoring
difficulty changes between different epochs. Next, we show an example where a
small miner can double-spend and generate the entire revenue from block
rewards in this model, assuming the other miners are honest, if the modified
fork-choice rule is used.

Consider the following setting. Suppose that the network starts with a genesis
block with difficulty 1, and in the next epoch, the difficulty is increased to
1000, and stays with that difficulty for $n$ additional epochs. A single miner
with $\frac{1}{500}$ of the total hashing power, can create a long range fork
from the genesis block, in which the difficulty does not increase. To ensure
that the attacker’s chain does not lag behind the others, in the last epoch,
the attacker publishes the timestamps at a slightly faster pace than the one
in the first honest epoch.

Note that the cumulative PoW in the honest chain is $1+1000\cdot n$ whereas
the attacker’s cumulative work is only $1+n$. Since the last epoch is created
slightly faster, the attacker has a longer chain, which would get selected by
the longest chain rule. This attack would not work with Bitcoin’s existing
fork-choice rule that chooses the chain with the most cumulative PoW, since
the attacker’s chain has only a tiny fraction of cumulative PoW compared to
the honest chain.


# On Briançon-Skoda theorem for foliations

Arturo Fernández-Pérez  Department of Mathematics. Federal University of Minas
Gerais. Av. Antônio Carlos, 6627 CEP 31270-901
Pampulha - Belo Horizonte - Brazil
<EMAIL_ADDRESS>,  Evelia R. García Barroso  Dpto. Matemáticas, Estadística
e Investigación Operativa
Sección de Matemáticas
Universidad de La Laguna. Apartado de Correos 456. 38200 La Laguna, Tenerife,
Spain<EMAIL_ADDRESS>and  Nancy Saravia-Molina  Dpto. Ciencias - Sección
Matemáticas, Pontificia Universidad Católica del Perú, Av. Universitaria 1801,
San Miguel, Lima 32, Peru<EMAIL_ADDRESS>
###### Abstract.

We generalize Mattei’s result relative to the Briançon-Skoda theorem for
foliations to the family of foliations of the second type. We use this
generalization to establish relationships between the Milnor and Tjurina
numbers of foliations of second type, inspired by the results obtained by Liu
for complex hypersurfaces and we determine a lower bound for the global
Tjurina number of an algebraic curve.

###### Key words and phrases:

holomorphic foliations, Briançon-Skoda theorem, Milnor number, Tjurina number.

###### 2010 Mathematics Subject Classification:

Primary 32S65 - Secondary 32S10

The first-named and third-named authors were partially supported by PUCP-PERU
dgi: 2020-3-0014. The first-named author was partially supported by FAPERJ-
Pronex E-26/010.001270/2016 and CNPq PQ-302790/2019-5. The second-named and
third-named authors were partially supported by the Pontificia Universidad
Católica del Perú project CAP 2019-ID 717. The first-named and second-named
author was supported by the grant PID2019-105896GB-I00 funded by
MCIN/AEI/10.13039/501100011033.

##  1\. Introduction and Statement of the results

The problem of deciding whether an element of a ring belongs to a given ideal
of the ring is known as the ideal membership and dates back to works of
Dedekind who gave the precised definition of an ideal. Even if we know
generators of the ideal, it is not trivial to determine if an element is a
member of it. Therefore it is interesting to give sufficient conditions for
ideal membership. An important theorem in this line is the Hilbert’s
Nullstellensatz: it states that if $I$ in an ideal in the ring of germs of
holomorphic functions at $0\in\mathbb{C}^{n}$ and $f$ vanishes on the zero
locus of $I$ then there is a power of $f$ belonging to $I$. The Briançon-Skoda
Theorem can be seen as an effective version of the Hilbert Nullstellensatz
when $I$ is a jacobian ideal. Let us clarify this last statement. Let
$f(x_{1},\ldots,x_{n})\in\mathbb{C}\\{x_{1},\ldots,x_{n}\\}$ be a non-unit
convergent power series. Consider its jacobian ideal
$J(f)=(\partial_{x_{1}}f,\ldots,\partial_{x_{n}}f)$. According to Wall [18] it
was Mather who asked about the smallest $r$ for which $f^{r}\in J(f)$. It was
known then that $f$ is an element of the integral closure of $J(f)$, which
implies the existence of a power of $f$ belonging to $J(f)$. At that time it
was also known, thanks to Saito [14], that if the origin is an isolated
critical point of $f$ then $f$ belongs to $J(f)$ iff $f$ is a quasi-
homogeneous polynomial. Briançon and Skoda [15] proved, using analytic results
of Skoda, that $f^{n}\in J(f)$. Later, Lipman and Teissier [8] gave an
algebraic proof of this algebraic statement. Subsequently, Briançon-Skoda
Theorem has been generalized in different contexts, and has given rise to
abundant literature. In Foliation Theory, Mattei proved

######  Theorem $1$.

([11, Théorème C]) Let $\mathcal{F}$ be a non-dicritical generalized curve
holomorphic foliation at $(\mathbb{C}^{2},p)$ given by
$\omega=P(x,y)dx+Q(x,y)dy$. If $f(x,y)=0$ is the reduced curve of total union
of separatrices of $\mathcal{F}$ then $f^{2}$ belongs to the ideal $(P,Q)$.

In this paper, we extend Theorem $1$ to the family of second type foliations
(perhaps dicritical) and show (see Example 3.3) that it is essential that the
foliation be of the second type.

######  Theorem A.

Let $\mathcal{F}$ be a germ of a second type holomorphic foliation at
$(\mathbb{C}^{2},p)$ induced by $\omega=P(x,y)dx+Q(x,y)dy$, where
$P,Q\in\mathbb{C}\\{x,y\\}$, and let $F=f/h$ be a reduced balanced equation of
separatrices for $\mathcal{F}$. Then $f^{2}$ belongs to ideal $(P,Q)$.

In Section 2 we introduce all the notions and tools necessary to prove Theorem
A. We are inspired by Mattei’s proof but to extend it to the dicritical case
we use the characterizations of the dicritical second type foliations given by
Genzmer in [6]. The proof of Theorem A is given in Section 3. In Section 4, we
obtain relationships between the Milnor number, $\mu_{p}(\mathcal{F})$, and
the Tjurina number, $\tau_{p}(\mathcal{F},\mathcal{B}_{0})$, of the foliation
$\mathcal{F}$ with respect to the zero divisor $\mathcal{B}_{0}$ of a balanced
divisor of separatrices $\mathcal{B}=\mathcal{B}_{0}-\mathcal{B}_{\infty}$ of
$\mathcal{F}$, inspiring us to do so in the work of Liu [9] for complex
hypersurfaces. More precisely, if $\mathcal{P}^{\mathcal{F}}$ is a generic
polar curve of $\mathcal{F}$, $\nu_{p}(.)$ denotes the algebraic multiplicity
of a curve and $i_{p}(.,.)$ denotes the intersection multiplicity of two
curves then we get

######  Theorem B.

Let $\mathcal{F}$ be a singular holomorphic foliation of second type at
$(\mathbb{C}^{2},p)$. Let $\mathcal{B}=\mathcal{B}_{0}-\mathcal{B}_{\infty}$
be a balanced divisor of separatrices for $\mathcal{F}$. Then

(1)
$\frac{(\nu_{p}(\mathcal{B}_{0})-1)^{2}+\nu_{p}(\mathcal{B}_{\infty})-i_{p}(\mathcal{P}^{\mathcal{F}},\mathcal{B}_{\infty})-i_{p}(\mathcal{B}_{0},\mathcal{B}_{\infty})}{2}\stackrel{{\scriptstyle(*)}}{{\leq}}\frac{\mu_{p}(\mathcal{F})}{2}\leq\tau_{p}(\mathcal{F},\mathcal{B}_{0}),$

and the equality $(*)$ holds if $\mathcal{F}$ is a generalized curve foliation
and $\mathcal{B}_{0}$ is defined by a germ of semi-homogeneous function at
$p$. Moreover, if $\mathcal{B}_{\infty}=\emptyset$, then

$\frac{\nu_{p}(\mathcal{F})^{2}}{2}\leq\frac{\mu_{p}(\mathcal{F})}{2}\leq\tau_{p}(\mathcal{F},\mathcal{B}_{0}).$

Finally, as consequence of Theorem B, in Section 5, we obtain a lower bound
for the global Tjurina number of an algebraic curve.

##  2\. Preliminaries

Let $\mathcal{F}$ be a germ of singular holomorphic foliation at
$(\mathbb{C}^{2},p)$, in local coordinates $(x,y)$ centered at $p$, the
foliation is given by a holomorphic 1-form

(2) $\omega=P(x,y)dx+Q(x,y)dy,$

or by its dual vector field

(3)
$v=-Q(x,y)\frac{\partial}{\partial{x}}+P(x,y)\frac{\partial}{\partial{y}},$

where $P(x,y),Q(x,y)\in{\mathbb{C}\\{x,y\\}}$ are relatively prime, where
$\mathbb{C}\\{x,y\\}$ is the ring of complex convergent power series in two
variables. The algebraic multiplicity of $\mathcal{F}$, denoted by
$\nu_{p}(\mathcal{F})$, is the minimum of the orders $\nu_{p}(P)$,
$\nu_{p}(Q)$ at $p$ of the coefficients of $\omega$.

We say that $C:f(x,y)=0$, with $f(x,y)\in\mathbb{C}\\{x,y\\}$, is an
$\mathcal{F}$-_invariant_ curve if

$\omega\wedge df=(f.h)dx\wedge dy,$

for some $h\in\mathbb{C}\\{x,y\\}$. A separatrix of $\mathcal{F}$ is an
irreducible $\mathcal{F}$-invariant curve. Denote by $Sep_{p}(\mathcal{F})$
the set of all separatrices of $\mathcal{F}$ through $p$. If
$Sep_{p}(\mathcal{F})$ is a finite set then we say that the foliation
$\mathcal{F}$ is non-dicritical and we call total union of separatrices of
$\mathcal{F}$ to the union of all elements of $Sep_{p}(\mathcal{F})$.
Otherwise we will say that $\mathcal{F}$ is a dicritical foliation.

A point $p\in\mathbb{C}^{2}$ is a reduced or simple singularity for
$\mathcal{F}$ if the linear part ${\rm D}v(p)$ of the vector field $v$ in (3)
is non-zero and has eigenvalues $\lambda_{1},\lambda_{2}\in\mathbb{C}$ fitting
in one of the two following cases:

  1. (1)

$\lambda_{1}\lambda_{2}\neq 0$ and
$\lambda_{1}/\lambda_{2}\not\in\mathbb{Q}^{+}$ (in which case we say that $p$
is a non-degenerate or complex hyperbolic singularity).

  2. (2)

$\lambda_{1}\neq 0$ and $\lambda_{2}=0$ (in which case we say that $p$ is a
saddle-node singularity).

The reduction process of the singularities of a codimension one singular
foliation over an ambient space of dimension two was achieved by Seidenberg
[16].

A singular foliation $\mathcal{F}$ at $(\mathbb{C}^{2},p)$ is a generalized
curve foliation if it has no saddle-nodes in its reduction process of
singularities. This concept was defined by Camacho-Lins Neto-Sad [3, Page
144]. In this case, there is a system of coordinates $(x,y)$ in which
$\mathcal{F}$ is induced by the equation

(4) $\omega=x(\lambda_{1}+a(x,y))dy-y(\lambda_{2}+b(x,y))dx,$

where $a(x,y),b(x,y)\in{\mathbb{C}}\\{x,y\\}$ are non-units, so that
$Sep_{p}(\mathcal{F})$ is formed by two transversal analytic branches given by
$\\{x=0\\}$ and $\\{y=0\\}$. In the case (2), up to a formal change of
coordinates, the saddle-node singularity is given by a 1-form of the type

(5) $\omega=x^{k+1}dy-y(1+\lambda x^{k})dx,$

where $\lambda\in\mathbb{C}$ and $k\in\mathbb{Z}_{>0}$ are invariants after
formal changes of coordinates (see [13, Proposition 4.3]). The curve
$\\{x=0\\}$ is an analytic separatrix, called strong, whereas $\\{y=0\\}$
corresponds to a possibly formal separatrix, called weak or central.

Let $\mathcal{F}$ be a foliation at $(\mathbb{C}^{2},p)$, given by a $1$-form
as in (2), with reduction process
$\pi:(\tilde{X},\mathcal{D})\to(\mathbb{C}^{2},p)$ and let
$\tilde{\mathcal{F}}=\pi^{*}\mathcal{F}$ be the strict transform of
$\mathcal{F}$. Denote by $\mathrm{Sing}(\cdot)$ the set of singularities of a
foliation. A saddle-node singularity $q\in\mathrm{Sing}(\tilde{\mathcal{F}})$
is said to be a tangent saddle-node if its weak separatrix is contained in the
exceptional divisor $\mathcal{D}$, that is, the weak separatrix is an
irreducible component of $\mathcal{D}$.

A foliation is in the second class or is of second type if there are no
tangent saddle-nodes in its reduction process of singularities. This notion
was studied by Mattei-Salem [12] in the non-dicritical case and by Genzmer [6]
for arbitrary foliations.

For a fixed reduction process of singularities
$\pi:(\tilde{X},\mathcal{D})\to(\mathbb{C}^{2},p)$ for $\mathcal{F}$, a
component $D\subset\mathcal{D}$ can be:

  * •

non-dicritical, if $D$ is $\tilde{\mathcal{F}}$-invariant. In this case, $D$
contains a finite number of simple singularities. Each non-corner singularity
of $D$ carries a separatrix transversal to $D$, whose projection by $\pi$ is a
curve in $Sep_{p}(\mathcal{F})$. Remember that a corner singularity of $D$ is
an intersection point of $D$ with other irreducible component of
$\mathcal{D}$.

  * •

dicritical, if $D$ is not $\tilde{\mathcal{F}}$-invariant. The reduction
process of singularities gives that $D$ may intersect only non-dicritical
components of $\mathcal{D}$ and $\tilde{\mathcal{F}}$ is everywhere transverse
to $D$. The $\pi$-image of a local leaf of $\tilde{\mathcal{F}}$ at each non-
corner point of $D$ belongs to $Sep_{p}(\mathcal{F})$.

Denote by $Sep_{p}(D)\subset Sep_{p}(\mathcal{F})$ the set of separatrices
whose strict transforms by $\pi$ intersect the component
$D\subset\mathcal{D}$. If $B\in Sep_{p}(D)$ with $D$ non-dicritical, $B$ is
said to be isolated. Otherwise, it is said to be a dicritical separatrix. This
determines the decomposition $Sep_{p}(\mathcal{F})=Iso_{p}(\mathcal{F})\cup
Dic_{p}(\mathcal{F})$, where notations are self-evident. The set
$Iso_{p}(\mathcal{F})$ is finite and contains all purely formal separatrices.
It subdivides further in two classes: weak separatrices — those arising from
the weak separatrices of saddle-nodes — and strong separatrices —
corresponding to strong separatrices of saddle-nodes and separatrices of non-
degenerate singularities. On the other hand, if $Dic_{p}(\mathcal{F})$ is non-
empty then it is an infinite set of analytic separatrices. Observe that a
foliation $\mathcal{F}$ is dicritical when $Sep_{p}(\mathcal{F})$ is infinite,
which is equivalent to saying that $Dic_{p}(\mathcal{F})$ is non-empty.
Otherwise, $\mathcal{F}$ is non-dicritical.

Throughout the text, we would rather adopt the language of divisors of formal
curves. More specifically, a divisor of separatrices for a foliation
$\mathcal{F}$ at $(\mathbb{C}^{2},p)$ is a formal sum

(6) $\mathcal{B}=\sum_{B\in\text{Sep}_{p}(\mathcal{F})}a_{B}\cdot B,$

where the coefficients $a_{B}\in\mathbb{Z}$ are zero except for finitely many
$B\in Sep_{p}(\mathcal{F})$. The set of separatrices $\\{B\;:\;a_{B}\neq 0\\}$
appearing in (6) is called the _support_ of the divisor $\mathcal{B}$ and it
is denoted by $\hbox{\rm supp}(\mathcal{B})$. The _degree_ of the divisor
$\mathcal{B}$ is by definition $\deg\mathcal{B}=\sum_{B\in\hbox{\rm
supp}(\mathcal{B})}a_{B}$. We denote by $Div_{p}(\mathcal{F})$ the set of all
these divisors of separatrices, which turns into a group with the canonical
additive structure. We follow the usual terminology and notation:

  * •

$\mathcal{B}\geq 0$ denotes an effective divisor, one whose coefficients are
all non-negative;

  * •

there is a unique decomposition
$\mathcal{B}=\mathcal{B}_{0}-\mathcal{B}_{\infty}$, where
$\mathcal{B}_{0},\mathcal{B}_{\infty}\geq 0$ are respectively the zero and
pole divisors of $\mathcal{B}$;

  * •

the algebraic multiplicity of $\mathcal{B}$ is
$\nu_{p}(\mathcal{B})=\displaystyle\sum_{B\in\hbox{\rm
supp}(\mathcal{B})}\nu_{p}(B).$

Following [6, page 5] and [7, Definition 3.1], we define a _balanced divisor
of separatrices_ for $\mathcal{F}$ as a divisor of the form

$\mathcal{B}\ =\ \sum_{B\in{\rm Iso}_{p}(\mathcal{F})}B+\sum_{B\in{\rm
Dic}_{p}(\mathcal{F})}\ a_{B}\cdot B,$

where the coefficients $a_{B}\in\mathbb{Z}$ are non-zero except for finitely
many $B\in Dic_{p}(\mathcal{F})$, and, for each dicritical component
$D\subset\mathcal{D}$, the following equality is respected:

$\sum_{B\in{\text{Dic}_{p}(D)}}a_{B}=2-val(D).$

The integer $val(D)$ stands for the valence of a component
$D\subset\mathcal{D}$ in the reduction process of singularities, that is, it
is the number of components of $\mathcal{D}$ intersecting $D$ other from $D$
itself.

The notion of balanced divisor of separatrices generalizes, to dicritical
foliations, the notion of total union of separatrices for non-dicritical
foliations.

A balanced divisor $\mathcal{B}=\sum_{B}a_{B}B$ of separatrices of
$\mathcal{F}$ is called primitive if, $a_{B}\in\\{-1,1\\}$ for any
$B\in\hbox{\rm supp}(\mathcal{B})$. A balanced equation of separatrices is a
formal meromorphic function ${F}(x,y)$ whose associated divisor
$\mathcal{B}=\mathcal{B}_{0}-\mathcal{B}_{\infty}$ is a balanced divisor. A
balanced equation is reduced or primitive if the same is true for the
underlying divisor.

By [6, Proposition 2.4] we have

(7) $\nu_{p}(\mathcal{F})=\nu_{p}(\mathcal{B})-1+\xi_{p}(\mathcal{F})$

and

(8) $\mathcal{F}$ is a second type foliation if, and only if,
$\;\nu_{p}(\mathcal{F})=\nu_{p}(\mathcal{B})-1,\;\;\;$

where $\mathcal{B}$ is a balanced divisor of separatrices for $\mathcal{F}$
and $\xi_{p}(\mathcal{F})$ is the tangency excess of $\mathcal{F}$ at $p$ (see
[5, Definition 2.3]).

##  3\. Proof of Theorem A

Let $\mathcal{F}$ be a germ of a singular holomorphic foliation at
$(\mathbb{C}^{2},p)$ induced by $\omega:=P(x,y)dx+Q(x,y)dy$, where
$P,Q\in\mathbb{C}\\{x,y\\}$, and consider a blow-up
$\sigma:(\tilde{\mathbb{C}}^{2},\tilde{D})\to(\mathbb{C}^{2},p)$ centered at
$p$ with $\tilde{D}=\sigma^{-1}(p)$ and let
$\tilde{\mathcal{F}}=\sigma^{*}(\mathcal{F})$ be the strict transform of
$\mathcal{F}$ by $\sigma$. Let $\mathcal{X}_{\tilde{\mathcal{F}}}$ be the
sheaf of (holomorphic) vector fields tangent to $\tilde{\mathcal{F}}$ and
$H^{1}(\tilde{D},\mathcal{X}_{\tilde{\mathcal{F}}})$ the first cohomology
group of $\mathcal{X}_{\tilde{\mathcal{F}}}$ on $\tilde{D}$·

We have the following lemma, which generalizes [10, Lemme 2.2.1] to dicritical
blow-ups.

######  Lemma 3.1.

With the above notation, we have

$\dim
H^{1}(\tilde{D},\mathcal{X}_{\tilde{\mathcal{F}}})=\frac{(\nu_{p}(\mathcal{F})-\epsilon_{p}(\mathcal{F})-1)(\nu_{p}(\mathcal{F})-\epsilon_{p}(\mathcal{F})-2)}{2},$

where

$\epsilon_{p}(\mathcal{F})=\begin{cases}0&\text{if $\sigma$ is non-
dicritical}\vskip 6.0pt plus 2.0pt minus 2.0pt\\\ 1&\text{if $\sigma$ is
dicritical}.\end{cases}$

###### Proof.

Let $x_{1}=x$, $y_{1}=\frac{y}{x}$ and $x_{2}=\frac{x}{y}$, $y_{2}=y$ be the
local coordinates of $\tilde{\mathbb{C}}^{2}$. Let
$V_{1}=\\{(0,y_{1}):|y_{1}|<2\\}$ and $V_{2}=\\{(x_{2},0):|x_{2}|<2\\}$, we
have $\tilde{D}=V_{1}\cup V_{2}$ and so that $\mathcal{V}=\\{V_{i}\\}_{i=1,2}$
is an open covering of $\tilde{D}$. Let
$v=-Q(x,y)\partial_{x}+P(x,y)\partial_{y}$ be the vector field defining
$\mathcal{F}$, up some calculations, we obtain that $\sigma^{*}(\mathcal{F})$
is given by the vector field $v^{(1)}$ satisfy over each chart of
$\tilde{\mathbb{C}}^{2}$:

(9)
$X_{1}=\frac{1}{x_{1}^{\nu_{p}(\mathcal{F})-\epsilon_{p}(\mathcal{F})}}\cdot
v^{(1)},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,X_{2}=\frac{1}{y_{2}^{\nu_{p}(\mathcal{F})-\epsilon_{p}(\mathcal{F})}}\cdot
v^{(1)}.$

Thus $\mathcal{X}_{\tilde{\mathcal{F}}}(V_{i})=\mathcal{O}(V_{i})\cdot X_{i}$
for each $i=1,2$. Set $V_{12}:=V_{1}\cap
V_{2}=\\{(0,y_{1}):\frac{1}{2}<|y_{1}|<2\\}$. Every section of
$\mathcal{X}_{\tilde{\mathcal{F}}}(V_{12})$ can be written as
$g(x_{1},y_{1})\cdot X_{1}$, where

(10)
$g(x_{1},y_{1})=\sum_{(i,j)\in\mathbb{N}\times\mathbb{Z}}g_{ij}x_{1}^{i}y_{1}^{j}$

is a convergent power series on

(11)
$\frac{1}{2}<|y_{1}|<2\,\,\,\,\,\text{and}\,\,\,\,\,\,\,|x_{1}|<\Gamma_{g}(y_{1}),$

for some continuous function $\Gamma_{g}:V_{12}\to\mathbb{R}_{+}$. Since
elements of $\mathcal{X}_{\tilde{\mathcal{F}}}(V_{1})$ can be generate with
series of type (10) whose terms $(i,j)$ with $j<0$ are all zeros, we can
consider
$\mathcal{X}_{\tilde{\mathcal{F}}}(V_{1})\subset\mathcal{X}_{\tilde{\mathcal{F}}}(V_{12})$.
With respect to $\mathcal{X}_{\tilde{\mathcal{F}}}(V_{2})$, their elements are
generated by convergent power series of the form

$k(x_{2},y_{2})=\sum_{(\alpha,\beta)\in\mathbb{N}^{2}}k_{\alpha\beta}x_{2}^{\alpha}y_{2}^{\beta}.$

Therefore, using (9), we get

$g(x_{1},y_{1})=\frac{1}{y^{\nu_{p}(\mathcal{F})-\epsilon_{p}(\mathcal{F})}_{1}}k(\frac{1}{y_{1}},x_{1}y_{1}),$

assuming the convergency of $k(x_{2},y_{2})$ on $\frac{1}{2}<|x_{2}|<2$ and
$|y_{2}|<\Gamma_{k}(x_{2})$, for some continuous function
$\Gamma_{k}:V_{12}\to\mathbb{R}_{+}$. Hence,
$\mathcal{X}_{\tilde{\mathcal{F}}}(V_{2})$ can be generated by power series of
type (10) whose terms $(i,j)$ with
$j>i-\nu_{p}(\mathcal{F})+\epsilon_{p}(\mathcal{F})$ are all zeros and so that
$\mathcal{X}_{\tilde{\mathcal{F}}}(V_{2})\subset\mathcal{X}_{\tilde{\mathcal{F}}}(V_{12})$.
Then, applying Leray’s theorem, we obtain

$H^{1}(\tilde{D},\mathcal{X}_{\tilde{\mathcal{F}}})=\check{H}^{1}(\mathcal{V},\mathcal{X}_{\tilde{\mathcal{F}}})=\frac{\mathcal{X}_{\tilde{\mathcal{F}}}(V_{12})}{\mathcal{X}_{\tilde{\mathcal{F}}}(V_{1})+\mathcal{X}_{\tilde{\mathcal{F}}}(V_{2})}.$

Thus, a basis for $H^{1}(\tilde{D},\mathcal{X}_{\tilde{\mathcal{F}}})$ is
given by the sections

(12)
$X_{ij}=x_{1}^{i}y_{1}^{j}X_{1}\in\mathcal{X}_{\tilde{\mathcal{F}}}(V_{12})\,\,\,\,\,\,\text{such
that}\,\,\,\,\,\,\,\,i\geq
0,\,\,\,\,\,\text{and}\,\,\,\,\,\,\,i-\nu_{p}(\mathcal{F})+\epsilon_{p}(\mathcal{F})<j<0.$

In particular, the dimension of
$H^{1}(\tilde{D},\mathcal{X}_{\tilde{\mathcal{F}}})$ is
$\frac{(\nu_{p}(\mathcal{F})-\epsilon_{p}(\mathcal{F})-1)(\nu_{p}(\mathcal{F})-\epsilon_{p}(\mathcal{F})-2)}{2}$.
∎

Let
$\pi:=\sigma_{1}\circ\ldots\circ\sigma_{\ell}:(\tilde{X},\mathcal{D})\to(\mathbb{C}^{2},p)$
be a reduction of singularities of $\mathcal{F}$ at $p\in\mathbb{C}^{2}$.
Denote by $F=f/h$ a reduced balanced equation of separatrices for
$\mathcal{F}$ and by $Z_{0}$ and $Z_{\infty}$ the respective strict transforms
by $\pi$ of the curves $\\{f=0\\}$ and $\\{h=0\\}$. Let
$\tilde{\mathcal{F}}=\pi^{*}(\mathcal{F})$ be the strict transform of
$\mathcal{F}$ by $\pi$. Let $\mathcal{X}_{\tilde{\mathcal{F}}}$ be the sheaf
of vector fields tangent to $\tilde{\mathcal{F}}$ and let
$\mathcal{X}_{Z_{0}}$ be the sheaf of vector fields tangent to the divisor
$\mathcal{D}$ and to $Z_{0}$.

######  Proposition 3.2.

Let $\mathcal{F}$ be a germ of a second type holomorphic foliation at
$(\mathbb{C}^{2},p)$ and let $F=f/h$ be a reduced balanced equation of
separatrices for $\mathcal{F}$. Put $\varphi=f\circ\pi$. Then, the morphism

$[\varphi]\cdot:H^{1}(\mathcal{D},\mathcal{X}_{\tilde{\mathcal{F}}})\to
H^{1}(\mathcal{D},\mathcal{X}_{\tilde{\mathcal{F}}}),\,\,\,\,\,\,[Y_{ij}]\longmapsto[\varphi\cdot
Y_{ij}]$

is identically zero.

###### Proof.

We will prove by induction on the number $\ell$ of blow-ups needed to obtain
the reduction of singularities of $\mathcal{F}$. If $\ell=1$, then
$H^{1}(\tilde{D},\mathcal{X}_{\tilde{\mathcal{F}}})$ is of finite dimension by
Lemma 3.1 and it follows from (12) that

$X_{ij}=x_{1}^{i}y_{1}^{j}X_{1}\,\,\,\,\,\,\text{such
that}\,\,\,\,\,\,\,\,(i,j)\in I=\\{(i,j):i\geq
0,\,\,\,\,\text{and}\,\,\,\,\,i-\nu_{p}(\mathcal{F})+\epsilon_{p}(\mathcal{F})<j<0\\}$

induce a basis for $H^{1}(\tilde{D},\mathcal{X}_{\tilde{\mathcal{F}}}),$ where
$x_{1}=x$, $y_{1}=\frac{y}{x}$ and $x_{2}=\frac{x}{y}$, $y_{2}=y$ are the
local coordinates of the blow-up $\tilde{X}$ and $X_{1}$ is as (9). Therefore
the sections of the form

(13) $x_{1}^{i}y_{1}^{j}X_{1}\,\,\,\,\,\,\text{such that}\,\,\,\,\,\,\,\,i\geq
0,\,\,\,\,j\geq 0\,\,\,\,\,\text{or}\,\,\,\,\,\,\,j\leq
i-\nu_{p}(\mathcal{F})+\epsilon_{p}(\mathcal{F})$

are elements of $\check{B}(\mathcal{D},\mathcal{X}_{\tilde{\mathcal{F}}})$
(i.e. 1-coboundary of $\mathcal{X}_{\tilde{\mathcal{F}}}$). Since
$\mathcal{F}$ is of second type, $\nu_{p}(\mathcal{F})=\nu_{p}(F)-1$, which
implies that $\nu_{p}(f)=\nu_{p}(h)+\nu_{p}(\mathcal{F})+1$. In particular,
$\varphi=f\circ\pi\in(x_{1}^{\nu_{p}(h)+\nu_{p}(\mathcal{F})+1})$. Hence the
sections $\varphi\cdot X_{ij}$ with $(i,j)\in I$ are elements of
$\check{B}(\mathcal{D},\mathcal{X}_{\tilde{\mathcal{F}}})$ and the proof of
proposition ends for $\ell=1$.

Now, for the general case, we use the exact sequence (see [10, page 312]):

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(\tilde{D},\mathcal{X}_{\tilde{\mathcal{F}}^{1}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{H^{1}(\mathcal{D},\mathcal{X}_{\tilde{\mathcal{F}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{H^{1}(D^{\prime},\mathcal{X}_{\tilde{\mathcal{F}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where $\tilde{D}=\sigma_{1}^{-1}(p)$, $\tilde{\mathcal{F}}^{1}$ is the strict
transform of $\mathcal{F}$ by $\sigma_{1}$, $D^{\prime}$ is the union of
irreducible components of $\mathcal{D}$ different of $\tilde{D}$, $\psi$ is
the restriction morphism and $\rho$ is the morphism induced by the natural
inclusion of $\tilde{D}$ in $\mathcal{D}$. Finally, since the following
diagram is commutative

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(\tilde{D},\mathcal{X}_{\tilde{\mathcal{F}}^{1}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\scriptstyle{[f\circ\sigma_{1}]\cdot}$$\textstyle{H^{1}(\mathcal{D},\mathcal{X}_{\tilde{\mathcal{F}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\scriptstyle{[\varphi]\cdot}$$\textstyle{H^{1}(D^{\prime},\mathcal{X}_{\tilde{\mathcal{F}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[f\circ\sigma_{2}\circ\ldots\circ\sigma_{\ell}]\cdot}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(\tilde{D},\mathcal{X}_{\tilde{\mathcal{F}}^{1}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{H^{1}(\mathcal{D},\mathcal{X}_{\tilde{\mathcal{F}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{H^{1}(D^{\prime},\mathcal{X}_{\tilde{\mathcal{F}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

we get $[\varphi]\cdot$ is identically zero, because $[f\circ\sigma_{1}]\cdot$
and $[f\circ\sigma_{2}\circ\ldots\circ\sigma_{\ell}]\cdot$ are morphisms
identically zero by the first step of the proof and induction hypothesis,
respectively. ∎

Now, we prove our main result.

######  Theorem A.

Let $\mathcal{F}$ be a germ of a second type holomorphic foliation at
$(\mathbb{C}^{2},p)$ induced by $\omega=P(x,y)dx+Q(x,y)dy$, where
$P,Q\in\mathbb{C}\\{x,y\\}$, and let $F=f/h$ be a reduced balanced equation of
separatrices for $\mathcal{F}$. Then $f^{2}$ belongs to ideal $(P,Q)$.

###### Proof.

Let $\pi:(\tilde{X},\mathcal{D})\to(\mathbb{C}^{2},p)$ be a reduction of
singularities of $\mathcal{F}$ and $\tilde{\mathcal{F}}=\pi^{*}(\mathcal{F})$
be the strict transform of $\mathcal{F}$ by $\pi$. According to Genzmer [6,
Proposition 3.1], since $\mathcal{F}$ is of second type, we have the exact
sequence of sheaves

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{X}_{\tilde{\mathcal{F}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{X}_{Z_{0}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{*}(\frac{\omega}{F})}$$\textstyle{\mathcal{O}(-Z_{\infty})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

where $\mathcal{X}_{\tilde{\mathcal{F}}}$ be the sheaf of vector fields
tangent to $\tilde{\mathcal{F}}$ and let $\mathcal{X}_{Z_{0}}$ be the sheaf of
vector fields tangent to the divisor $\mathcal{D}$ and to $Z_{0}$. Then, there
exists a covering of $\mathcal{D}$ by open subsets $V_{i}\subset\tilde{X}$ and
holomorphic vector fields $X_{i}\in\mathcal{X}_{Z_{0}}(V_{i})$ such that

$\pi^{*}\left(\frac{\omega}{F}\right)(X_{i})=h\circ\pi,\,\,\,\,\,\,\,\mathcal{O}(-Z_{\infty})=(h\circ\pi)\mathcal{O},$

which implies that

(14) $\pi^{*}(\omega)(X_{i})=(F\circ\pi)\cdot(h\circ\pi)=f\circ\pi.$

Let $X_{ij}:=X_{i}-X_{j}$. It follows from (14) that
$\pi^{*}(\omega)(X_{ij})=0$. Hence $X_{ij}$ is a 1-cocycle with values over
the sheaf $\mathcal{X}_{\tilde{\mathcal{F}}}$ and therefore

$[(f\circ\pi)X_{ij}]=0\in
H^{1}(\mathcal{D},\mathcal{X}_{\tilde{\mathcal{F}}}),$

by Proposition 3.2. Thus, there exists a holomorphic vector field $\tilde{v}$
on $\mathcal{D}$ such that $\tilde{v}|_{V_{i}}=(f\circ\pi)\cdot X_{i}$. Up
multiplication by $f\circ\pi$ in (14), we get

$\pi^{*}(\omega)(\tilde{v})=(f\circ\pi)^{2}=f^{2}\circ\pi.$

The direct image of $\tilde{v}$ by $\pi$ over $(\mathbb{C}^{2},p)$ is a
holomorphic vector field outside the origin of $\mathbb{C}^{2}$. The proof
ends, by applying Hartogs extension theorem.∎

We note that Theorem A is optimal, in the sense, that the hypothesis on the
foliation be of second type cannot be removed. For instance, we have the
following example.

######  Example 3.3.

Let
$\omega=y(2x^{8}+2(\lambda+1)x^{2}y^{3}-y^{4})dx+x(y^{4}-(\lambda+1)x^{2}y^{3}-x^{8})dy$
be a $1$-form defining a singular foliation $\mathcal{F}$ at
$(\mathbb{C}^{2},0)$, which is not of second type and $xy=0$ is the equation
of an effective divisor of separatrices for $\mathcal{F}$ (see [5, Example
6.5]). We claim that $(xy)^{2}$ does not belong to the ideal generated by the
components of $\omega$. In fact, if
$P(x,y):=y(2x^{8}+2(\lambda+1)x^{2}y^{3}-y^{4})$,
$Q(x,y):=x(y^{4}-(\lambda+1)x^{2}y^{3}-x^{8})$ and we suppose that
$(xy)^{2}=a(x,y)P(x,y)+b(x,y)Q(x,y)$ for some
$a(x,y),b(x,y)\in\mathbb{C}[[x,y]]$ then
$4=\text{ord}(xy)^{2}\geq\min\\{\text{ord}(a(x,y)P(x,y),\text{ord}(b(x,y)Q(x,y)\\}\geq
5$ which is a contradiction.

The following corollary will be useful in the following section:

######  Corollary 3.4.

Let $\mathcal{F}$ be a germ of a second type holomorphic foliation at
$(\mathbb{C}^{2},p)$ induced by $\omega=P(x,y)dx+Q(x,y)dy$, where
$P,Q\in\mathbb{C}\\{x,y\\}$, and let $\mathcal{B}$ be a reduced balanced
equation of separatrices for $\mathcal{F}$. If $\mathcal{B}_{0}:f(x,y)=0$ and
$\bar{f}$ is the coset of $f$ modulo $(P,Q)$ then the complex vector spaces
$(f,P,Q)/(P,Q)$ and $(\bar{f})/(\bar{f}^{2})$ are isomorphic.

###### Proof.

Put $\mathfrak{T}=(f,P,Q)$. The map
$\psi:\mathfrak{T}\longrightarrow(\bar{f})/(\bar{f}^{2})$ given by

$\psi(g_{z}f+\alpha P+\beta Q)=\overline{g_{z}f}\;\;\hbox{\rm mod
}(\bar{f}^{2})$

is an epimorphism of complex vector spaces. Finally by Theorem A the kernel of
$\psi$ equals $(P,Q)$.

∎

##  4\. Milnor and Tjurina numbers after the Briançon-Skoda theorem

Let $\mathcal{F}$ be a singular holomorphic foliation at $(\mathbb{C}^{2},p)$
given by the $1$-form $\omega:=P(x,y)dx+Q(x,y)dy$. Assume that $\mathcal{F}$
has an isolated singularity at $p$ and consider the jacobian ideal associated
with $\mathcal{F}$ given by $J(\mathcal{F})=(P,Q)$. Then
$\mathcal{M}(\mathcal{F}):=\mathbb{C}[[x,y]]/J(\mathcal{F})$ is a finite
$\mathbb{C}$-dimensional vector space which dimension is called the Milnor
number of $\mathcal{F}$ and we denote it by $\mu_{p}(\mathcal{F})$. It is
well-known, after [3], that the Milnor number is a topological invariant of
the foliation. Let $C:f(x,y)=0$ be an $\mathcal{F}-$invariant reduced curve.
Put $\mathcal{T}(\mathcal{F},C):=\mathbb{C}[[x,y]]/(f,P,Q)$, where
$(\cdot,\cdot,\cdot)$ denotes the ideal generated by three elements in
$\mathbb{C}[[x,y]]$.

The Tjurina number of $\mathcal{F}$ with respect to $C$ is

$\tau_{p}(\mathcal{F},C)=\dim_{\mathbb{C}}\mathcal{T}(\mathcal{F},C).$

Let $\mathcal{B}$ be a balanced divisor of separatrices for $\mathcal{F}$. Put
$\mathcal{B}_{0}:f(x,y)=0$ the zero divisor of $\mathcal{B}$. By definition
$\tau_{p}(\mathcal{F},\mathcal{B}_{0})\leq\mu_{p}(\mathcal{F})$. Put
$\mathfrak{T}=(f,P,Q)$. From the third isomorphic theorem for complex vector
spaces we have

$\tau_{p}(\mathcal{F},\mathcal{B}_{0})=\dim_{\mathbb{C}}\mathbb{C}[[x,y]]/\mathfrak{T}=\dim_{\mathbb{C}}\mathcal{M}(\mathcal{F})-\dim_{\mathbb{C}}\mathfrak{T}/J(\mathcal{F}),$

so

(15)
$\mu_{p}(\mathcal{F})-\tau_{p}(\mathcal{F},\mathcal{B}_{0})=\dim_{\mathbb{C}}\mathfrak{T}/J(\mathcal{F}).$

For any $z\in\mathbb{C}[[x,y]]$ we denote by $\bar{z}$ the coset of $z$ modulo
$J(\mathcal{F})$ and $\hat{z}$ its coset modulo $\mathfrak{T}$. Inspired by
Liu [9] we consider the exact sequence

$0\longrightarrow{\rm Ker}\;\sigma\stackrel{{\scriptstyle
i}}{{\longrightarrow}}\mathcal{M}(\mathcal{F})\stackrel{{\scriptstyle\sigma}}{{\longrightarrow}}\mathcal{M}(\mathcal{F})\stackrel{{\scriptstyle\delta_{\mathcal{B}}}}{{\longrightarrow}}\mathcal{T}(\mathcal{F},\mathcal{B}_{0})\longrightarrow
0,$

where $i$ is the inclusion map, $\sigma$ is the multiplication by $\bar{f}$,
that is, $\sigma(\bar{z})=\overline{zf}$ and
$\delta_{\mathcal{B}}(\bar{z})=\hat{z}$. Since $\delta_{\mathcal{B}}$ is
surjective, we get

(16)
$\mu_{p}(\mathcal{F})-\tau_{p}(\mathcal{F},\mathcal{B}_{0})=\dim_{\mathbb{C}}{\rm
Ker}\;\delta_{\mathcal{B}}.$

From (16) and the equality $\mu_{p}(\mathcal{F})=\dim_{\mathbb{C}}{\rm
Ker}\;\sigma+\dim_{\mathbb{C}}{\rm Im}\;\sigma$, we conclude

(17) $\tau_{p}(\mathcal{F},\mathcal{B}_{0})=\dim_{\mathbb{C}}{\rm
Ker}\;\sigma=\dim_{\mathbb{C}}(J(\mathcal{F}):\mathcal{B}_{0})/J(\mathcal{F}),$

where $(J(\mathcal{F}):\mathcal{B}_{0})=\\{z\in\mathbb{C}[[x,y]]\;:\;zf\in
J(\mathcal{F})\\}$.

######  Proposition 4.1.

Let $\mathcal{F}$ be a singular holomorphic foliation of second type at
$(\mathbb{C}^{2},p)$ given by the $1$-form $\omega=P(x,y)dx+Q(x,y)dy=0$. Let
$\mathcal{B}$ be a balanced divisor of separatrices for $\mathcal{F}$ with
$\mathcal{B}_{0}:f(x,y)=0$. Then
$\tau_{p}(\mathcal{F},\mathcal{B}_{0})\leq\mu_{p}(\mathcal{F})\leq
2\tau_{p}(\mathcal{F},\mathcal{B}_{0})$. Moreover
$\mu_{p}(\mathcal{F})=2\tau_{p}(\mathcal{F},\mathcal{B}_{0})$ if and only if
$\ker\sigma=(\bar{f})$, where $\bar{f}$ is the coset of $f$ modulo $(P,Q)$.

###### Proof.

Let us prove the inequality $\mu_{p}(\mathcal{F})\leq
2\tau_{p}(\mathcal{F},\mathcal{B}_{0}).$ By Theorem A we get $f^{2}\in
J(\mathcal{F})$, that is,
$\overline{f^{2}}=\bar{0}\in\mathcal{M}(\mathcal{F})$. Hence, we get the
inclusion of ideals $\mathfrak{T}\subseteq{\rm Ker}\;\sigma$. Moreover we have
the following chain of ideals of $\mathcal{M}(\mathcal{F})$:

$\mathcal{M}(\mathcal{F})\supseteq(\bar{f})\supseteq(\bar{f^{2}})=(\bar{0})$

where $(\cdot)$ denotes a principal ideal. We also have the exact sequence:

$0\rightarrow{\rm Ker}\;\sigma\cap(\bar{f})\stackrel{{\scriptstyle
i}}{{\rightarrow}}(\bar{f})\stackrel{{\scriptstyle\sigma^{\prime}}}{{\rightarrow}}(\bar{f})\stackrel{{\scriptstyle
e}}{{\rightarrow}}(\bar{f})/(\bar{f^{2}})\rightarrow 0,$

where $i$ is the inclusion map, $\sigma^{\prime}$ is the multiplication by the
coset $\bar{f}$ and $e$ is the natural epimorphism. We have

$\displaystyle\dim_{\mathbb{C}}{\rm
Ker}\;\sigma^{\prime}+\dim_{\mathbb{C}}{\rm Im}\;\sigma^{\prime}$
$\displaystyle=$
$\displaystyle\dim_{\mathbb{C}}(\bar{f})=\dim_{\mathbb{C}}{\rm
Ker}\;e+\dim_{\mathbb{C}}{\rm Im}\;e$ $\displaystyle=$
$\displaystyle\dim_{\mathbb{C}}{\rm
Im}\;\sigma^{\prime}+\dim_{\mathbb{C}}(\bar{f})/(\bar{f^{2}}),$

so from (17) we get

$\dim_{\mathbb{C}}(\bar{f})/\bar{(f^{2}})=\dim_{\mathbb{C}}{\rm
Ker}\;\sigma^{\prime}=\dim_{\mathbb{C}}{\rm
Ker}\;\sigma\cap(\bar{f})\leq\dim_{\mathbb{C}}{\rm
Ker}\;\sigma=\tau(\mathcal{F},\mathcal{B}_{0}).$

After Corollary 3.4 we have
$\dim_{\mathbb{C}}(\bar{f})/(\bar{f}^{2})=\dim_{\mathbb{C}}\mathfrak{T}/J(\mathcal{F})$
and by (15) we conclude $\mu_{p}(\mathcal{F})\leq
2\tau_{p}(\mathcal{F},\mathcal{B}_{0}).$ Finally
$\mu_{p}(\mathcal{F})=2\tau_{p}(\mathcal{F},\mathcal{B}_{0})$ if and only if
${\rm Ker}\;\sigma\cap(\bar{f})={\rm Ker}\;\sigma$, so ${\rm
Ker}\;\sigma\subseteq(\bar{f})$. We conclude the proof since
$\sigma(\bar{f})=\bar{0}$. ∎

The intersection multiplicity of two curves $C:f(x,y)=0$ and $D:g(x,y)=0$ at
the point $p$ is by definition
$i_{p}(C,D)=\dim_{\mathbb{C}}\mathbb{C}\\{x,y\\}/(f,g)$ where $(f,g)$ denotes
the ideal of $\mathbb{C}\\{x,y\\}$ generated by the power series $f$ and $g$.

The polar curve of the singular foliation
$\mathcal{F}:\omega=P(x,y)dx+Q(x,y)dy=0$ at $(\mathbb{C}^{2},p)$ with respect
to a point $(a:b)$ of the complex projective line $\mathbb{P}^{1}(\mathbb{C})$
is the analytic curve $\mathcal{P}^{\mathcal{F}}_{(a:b)}:aP(x,y)+bQ(x,y)=0$.
There exists an open Zariski set $U$ of $\mathbb{P}^{1}(\mathbb{C})$ such that
$\\{aP(x,y)+bQ(x,y)=0\;:\;(a:b)\in U\\}$ is an equisingular family of plane
curves. Any element of this set is called generic polar curve of the foliation
$\mathcal{F}$ and we will denote it by $\mathcal{P}^{\mathcal{F}}$.

A germ of plane curve $C:f(x,y)=0$ of multiplicity $n$ is a semi-homogenous
function at $p$ if and only if $f=f_{n}+g$ where $f_{n}$ is a homogeneous
polinomial of degree $n$ defining an isolated singularity at $p$ and $g$
consists of terms of degree at least $n+1$.

######  Theorem B.

Let $\mathcal{F}$ be a singular holomorphic foliation of second type at
$(\mathbb{C}^{2},p)$. Let $\mathcal{B}=\mathcal{B}_{0}-\mathcal{B}_{\infty}$
be a balanced divisor of separatrices for $\mathcal{F}$. Then

(18)
$\frac{(\nu_{p}(\mathcal{B}_{0})-1)^{2}+\nu_{p}(\mathcal{B}_{\infty})-i_{p}(\mathcal{P}^{\mathcal{F}},\mathcal{B}_{\infty})-i_{p}(\mathcal{B}_{0},\mathcal{B}_{\infty})}{2}\stackrel{{\scriptstyle(*)}}{{\leq}}\frac{\mu_{p}(\mathcal{F})}{2}\leq\tau_{p}(\mathcal{F},\mathcal{B}_{0}),$

and the equality $(*)$ holds if $\mathcal{F}$ is a generalized curve foliation
and $\mathcal{B}_{0}$ is defined by a germ of semi-homogeneous function at
$p$. Moreover, if $\mathcal{B}_{\infty}=\emptyset$, then

$\frac{\nu_{p}(\mathcal{F})^{2}}{2}\leq\frac{\mu_{p}(\mathcal{F})}{2}\leq\tau_{p}(\mathcal{F},\mathcal{B}_{0}).$

###### Proof.

By [5, Proposition 4.2], for any singular foliation $\mathcal{F}$ we have

(19)
$\displaystyle\Delta_{p}(\mathcal{F},\mathcal{B}_{0})=i_{p}(\mathcal{P}^{\mathcal{F}},\mathcal{B}_{0})+i_{p}(\mathcal{B}_{0},\mathcal{B}_{\infty})-\mu_{p}(\mathcal{B}_{0})-\nu_{p}(\mathcal{B}_{0})+1,$

where $\Delta_{p}(\mathcal{F},\mathcal{B}_{0})$ is the excess polar number of
$\mathcal{F}$ with respect to $\mathcal{B}_{0}$. Since $\mathcal{F}$ is of
second type,
$\nu_{p}(\mathcal{F})=\nu_{p}(\mathcal{B})-1=\nu_{p}(\mathcal{B}_{0})-\nu_{p}(\mathcal{B}_{\infty})-1$
by equation (8), and therefore, from (19) we get

(20)
$\displaystyle\Delta_{p}(\mathcal{F},\mathcal{B}_{0})=i_{p}(\mathcal{P}^{\mathcal{F}},\mathcal{B}_{0})+i_{p}(\mathcal{B}_{0},\mathcal{B}_{\infty})-\mu_{p}(\mathcal{B}_{0})-\nu_{p}(\mathcal{F})-\nu_{p}(\mathcal{B}_{\infty}).$

On the other hand, after [7, Theorem A] we know that
$\Delta_{p}(\mathcal{F},\mathcal{B}_{0})\geq 0$, and equals zero if and only
if $\mathcal{F}$ is a generalized curve foliation. Hence from (20) we have

(21) $\displaystyle\mu_{p}(\mathcal{B}_{0})\leq
i_{p}(\mathcal{P}^{\mathcal{F}},\mathcal{B}_{0})+i_{p}(\mathcal{B}_{0},\mathcal{B}_{\infty})-\nu_{p}(\mathcal{F})-\nu_{p}(\mathcal{B}_{\infty}).$

Now, by applying [5, Lemma 4.4] to $\mathcal{F}$, which is of second type, and
by properties of the intersection multiplicity one gets

(22) $\displaystyle
i_{p}(\mathcal{P}^{\mathcal{F}},\mathcal{B}_{0})=i_{p}(\mathcal{P}^{\mathcal{F}},\mathcal{B}_{\infty})+\mu_{p}(\mathcal{F})+\nu_{p}(\mathcal{F}),$

so from (21) and (22),

(23)
$\displaystyle\mu_{p}(\mathcal{B}_{0})\leq\mu_{p}(\mathcal{F})+i_{p}(\mathcal{B}_{0},\mathcal{B}_{\infty})+i_{p}(\mathcal{P}^{\mathcal{F}},\mathcal{B}_{\infty})-\nu_{p}(\mathcal{B}_{\infty}).$

It follows from the definition of the Milnor number, the properties of the
intersection multiplicity and (23) that

(24)
$\displaystyle(\nu_{p}(\mathcal{B}_{0})-1)^{2}\leq\mu_{p}(\mathcal{B}_{0})\leq\mu_{p}(\mathcal{F})+i_{p}(\mathcal{B}_{0},\mathcal{B}_{\infty})+i_{p}(\mathcal{P}^{\mathcal{F}},\mathcal{B}_{\infty})-\nu_{p}(\mathcal{B}_{\infty}).$

Observe that the first inequality becomes an equality when $B_{0}$ is defined
by a germ of semi-homogeneous function at $p$ (see [17]) and the second
inequality is an equality if and only if $\mathcal{F}$ is a generalized curve
foliation. Finally, the proof ends, up applying Proposition 4.1

(25)
$(\nu_{p}(\mathcal{B}_{0})-1)^{2}+\nu_{p}(\mathcal{B}_{\infty})-i_{p}(\mathcal{B}_{0},\mathcal{B}_{\infty})-i_{p}(\mathcal{P}^{\mathcal{F}},\mathcal{B}_{\infty})\leq\mu_{p}(\mathcal{F})\leq
2\tau_{p}(\mathcal{F},\mathcal{B}_{0}).$

∎

######  Example 4.2.

We illustrate Theorem B with the radial foliation $\mathcal{F}$ given by the
$1$-form $\omega=xdy-ydx$. In this case we consider $\mathcal{B}_{0}=xy(x-y)$
and $\mathcal{B}_{\infty}=x+y.$ We get $\nu_{0}(\mathcal{B}_{0})=3$,
$1=\nu_{0}(\mathcal{B}_{\infty})=i_{0}(\mathcal{P}^{\mathcal{F}},\mathcal{B}_{\infty})=\tau_{0}(\mathcal{F},\mathcal{B}_{0})$
and $i_{0}(\mathcal{B}_{0},\mathcal{B}_{\infty})=3$. Hence $\mathcal{F}$
verifies (18).

######  Remark 4.3.

The family of foliations given in [5, Example 6.5] are defined by the $1$-form

$\omega_{k}=y(2x^{2k-2}+2(\lambda+1)x^{2}y^{k-2}-y^{k-1})dx+x(y^{k-1}-(\lambda+1)x^{2}y^{k-2}-x^{2k-2})dy$

is a family of dicritical foliations which are not of second type,
$\mathcal{B}=B_{1}+B_{2}$ is an effective balanced divisor of separatrices for
$\mathcal{F}_{k}$. We get $\nu_{0}(\mathcal{F}_{k})=k$ and
$\tau_{0}(\mathcal{F}_{k},\mathcal{B})=3k-2$. Hence the inequality

$\frac{\nu_{p}(\mathcal{F})^{2}}{2}\leq\tau_{p}(\mathcal{F},\mathcal{B})$

fails for all $k\geq 6$. Therefore, in Theorem B the second type hypothesis
over $\mathcal{F}$ is essential.

##  5\. A lower bound for the global Tjurina number of an algebraic curve

Let $C$ be a reduced curve of degree $\deg(C)$ in the complex projective plane
$\mathbb{P}^{2}$. Denote by $\tau(C)$ the global Tjurina number of the curve
$C$, which is the sum of the Tjurina numbers at the singular points of $C$. In
this section, under some conditions, we give a lower bound for $\tau(C)$.

A holomorphic foliation $\mathcal{F}$ on $\mathbb{P}^{2}$ of degree $d\geq 0$
is a foliation defined by a polynomial 1-form
$\Omega=A(x,y,z)dx+B(x,y,z)dy+C(x,y,z)dz,$ where $A,B,C$ are complex
homogeneous polynomials of degree $d+1$, satisfying two conditions:

  1. (1)

the integrability condition $\Omega\wedge d\Omega=0$,

  2. (2)

the Euler condition $Ax+By+Cz=0$.

An algebraic curve $C:f(x,y,z)=0$ is $\mathcal{F}$-invariant if $\Omega\wedge
df=f\Theta,$ where $\Theta$ is some polynomial 2-form.

Denote by $\left\lceil z\right\rceil$ the ceiling function evaluated at
$z\in\mathbb{R}$, that is, the smallest integer that is greater than or equal
to $z\in\mathbb{R}$. We have:

######  Theorem 5.1.

Let $\mathcal{F}$ be a holomorphic foliation on $\mathbb{P}^{2}$ of degree
$d$. Suppose that all points $p\in Sing(\mathcal{F})$ are of second type. Then

(26) $\left\lceil\frac{d^{2}+d+1-2\sum_{p\in
Sing(\mathcal{F})}GSV_{p}(\mathcal{F},(F_{p})_{0})}{2}\right\rceil\leq\sum_{p\in
Sing(\mathcal{F})}\tau_{p}((F_{p})_{0}),$

where $(F_{p})_{0}$ is the zero divisor of a balanced equation of separatrices
$F_{p}$ for $\mathcal{F}$ at $p$. In particular, if $C$ is an
$\mathcal{F}$-invariant reduced curve in $\mathbb{P}^{2}$ such that
$Sing(\mathcal{F})\subset C$ and for all $p\in Sing(\mathcal{F})$, the germ of
$C$ at $p$ defines the zero divisor of a balanced equation of separatrices for
$\mathcal{F}$ at $p$, then

(27)
$\left\lceil\frac{d^{2}+d+1-2(d+2)\deg(C)+2\deg(C)^{2}}{2}\right\rceil\leq\tau(C),$

###### Proof.

Since all points $p\in Sing(\mathcal{F})$ are of second type, then

(28) $\mu_{p}(\mathcal{F})\leq 2\tau_{p}(\mathcal{F},(F_{p})_{0})$

by Theorem B. According to [5, Proposition 6.2 ], we have
$\tau_{p}(\mathcal{F},(F_{p})_{0})=GSV_{p}(\mathcal{F},(F_{p})_{0})+\tau_{p}((F_{p})_{0})$.
Hence, up substituting in (28), we obtain

$\frac{\mu_{p}(\mathcal{F})-2GSV_{p}(\mathcal{F},(F_{p})_{0})}{2}\leq\tau_{p}((F_{p})_{0}),\,\,\,\,\,\,\text{for
all}\,\,\,p\in Sing(\mathcal{F}).$

The inequality (26) is proved by taking sum over all singular points of
$\mathcal{F}$, by using $\sum_{p\in
Sing(\mathcal{F})}\mu_{p}(\mathcal{F})=d^{2}+d+1$ (see [2, Page 28]) and
considering the ceiling function. The inequality (27) follows from

$\sum_{p\in Sing(\mathcal{F})\cap
C}GSV_{p}(\mathcal{F},C)=(d+2)\deg(C)-\deg(C)^{2}$

given in [1, Proposition 4] and considering again the ceiling function. ∎

The following example illustrates Theorem 5.1.

######  Example 5.2.

For each $\lambda\in\mathbb{C}$, we consider the 1-form

$\omega_{\lambda}=yzdx+\lambda xzdy-(\lambda+1)xydz,$

which defines a foliation $\mathcal{F}_{\lambda}$ on $\mathbb{P}^{2}$ of
degree one. The curve $C:$ $xyz=0$ has degree three and it satisfies all
hypotheses of Theorem 5.1. Then

$\left\lceil\frac{1^{2}+1+1-2(1+2)3+2\cdot
3^{2}}{2}\right\rceil=\left\lceil\frac{3}{2}\right\rceil=2\leq\tau(C)=3,$

which implies that the inequality (27) of Theorem 5.1 is verified. Observed
that we equate the bound given by du Plessis and Wall in [4, Theorem 3.2].

Acknowledgement. The first-named author thanks Universidad de La Laguna for
the hospitality during his visit in June 2022.

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