A Note on the Multiplicity of the Distinguished Points

Weiping Li; Xiaoshen Wang

doi:10.3390/axioms13070479

and

¹

Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, USA

²

Department of Mathematics and Statistics, University of Arkansas at Little Rock, Little Rock, AR 72204, USA

^*

Authors to whom correspondence should be addressed.

Axioms2024, 13(7), 479;https://doi.org/10.3390/axioms13070479

Version Notes

Order Reprints

Abstract

Let

P (x)

be a system of polynomials in s variables, where

x \in C^{s}

. If

z_{0}

is an isolated zero of P, then the multiplicity and its structure at

z_{0}

can be revealed by the normal set of the quotient ring

R (< P >)

or its dual space

R^{*}

or by certain numerical methods. In his book titled “Numerical Polynomial Algebra”, Stetter described the so-called distinguished points, which are embedded in a zero manifold of P, and the author defined their multiplicities. In this note, we will generalize the definition of distinguished points and give a more appropriate definition for their multiplicity, as well as show how to calculate the multiplicity of these points.

Keywords:

multiplicity; distinguished points; Puiseux series

MSC:

55-XX

Let

P (x)

be a system of polynomials in s variables, where

x \in C^{s}

. If

z_{0}

is an isolated zero of P, then the multiplicity and its structure at

z_{0}

can be revealed by the normal set of the quotient ring

R (< P >)

or its dual space

R^{*}

(cf. [1]), and a numerical method has been developed by Dayton and Zeng [2]. In [1], Stetter described the so-called distinguished points, which are embedded in a zero manifold of P, and defined their multiplicity. However, it turns out that in most of the cases, the points that display some kind of “distinguished” property do not satisfy the definitions for distinguished points. In addition, the multiplicity defined there is not quite appropriate. So in this note, we will generalize the definition of distinguished points and give a more appropriate definition for their multiplicity, as well as show how to calculate the multiplicity of these points.

Example 1.

For a polynomial f in one variable, if

f (x) = {(x - a)}^{m} q (x)

, where

q (a) \neq 0

, then a is a zero of f with multiplicity m. Equivalently, if there exists a maximal set of linearly independent linear functionals

{c_{1}, \dots, c_{m}}

evaluating at a such that

c_{i} (f) = 0

and

c_{i} (g f) = 0

for all g, then the multiplicity of f at a is m. An obvious choice of a set of such functionals is

c_{i} = \frac{d^{i - 1}}{(i - 1)! d x^{i - 1}}, i = 1, \dots, m

evaluating at

x = a

.

Now, we consider a perturbed equation

H (x, t) = f (x) - t = 0,

where t is a complex parameter. It is not too hard to see that the solutions to the equation can be expanded as a Puiseux series in t:

x_{i} (t) = a + u_{i} t^{(1 / m)} / q {(a)}^{1 / m} + h i g h e r o r d e r t e r m s

,

i = 1, \dots, m

, where

u_{i}

is an mth primitive root of unity converging to

x = a

. Define

d_{1} (f) = f (x_{1}) = f [x_{1}]

,

d_{2} (f) = f [x_{1}, x_{2}] = (f [x_{2}] - f [x_{1}]) / (x_{2} - x_{1}), \dots ., d_{m - 1} = (f [x_{1}, \dots, x_{m - 1}] - f [x_{2}, \dots,

x_{m}]) / (x_{m} - x_{1})

, where

f [x_{1}, \dots, x_{k + 1}]

denotes the standard Newton’s kth divided difference. Apparently, the

d_{i} s

depend on t. When

t \to 0

, the

d_{i} s

“converge” to

c_{i} s

.

On the other hand, if the zero set of the perturbed polynomial

H (x, t)

can be expanded as a Puiseux series in t—

x = x_{i} (t), i = 1, \dots, m

near

x = a

—then the multiplicity of a is m, since this means that for each

0 < | t | < < 1

,

H (x, t)

has exactly m zeros.

For polynomials with more variables, the definition above can be generalized. The Puiseux series has long been used to derive the bound of the number of zeros for the system of polynomial equations [3,4]. We will show that the Puiseux series can be used to reveal the multiplicity of a point isolated or not.

Example 2.

Suppose that

P (x, y) = {(x^{2}, y^{2})}^{t}

, where

(x, y) \in C^{2}

. It can be shown that

z_{0} = (0, 0)

is a zero of P with a multiplicity of four. Let

P_{1} (x, y) = - (\begin{matrix} c_{10} + c_{11} x + c_{12} y + \dots \\ c_{20} + c_{21} x + c_{22} y + \dots \end{matrix})

be a generic polynomial and define a perturbed system

Q_{t} : = P (x, y) + t P_{1} (x, y)

. Then for small t, there are four zeros of

P_{t}

near

z_{0}

. To see that, we substitute

x = x_{0} t^{α_{1}} + x_{1} t^{2 α_{1}} + \dots,

and

y = y_{0} t^{α_{2}} + y_{1} t^{2 α_{2}} + \dots,

for x and y in the following equations:

P (x, y) + t P_{1} (x, y) = 0

\begin{matrix} x_{0}^{2} t^{2 α_{1}} + \dots & = & t (c_{10} + c_{11} x_{0} t^{α_{1}} + c_{12} y_{0} t^{α_{2}} + \dots), \\ y_{0}^{2} t^{2 α_{2}} + \dots & = & t (c_{20} + c_{21} x_{0} t^{α_{2}} + c_{22} y_{0} t^{α_{2}} + \dots) . \end{matrix}

Equating the coefficients and the exponents of the lowest order terms in t yields

\begin{matrix} x_{0}^{2} & = & c_{10}, \\ y_{0}^{2} & = & c_{20}, \\ 2 α_{1} & = & 1, \\ 2 α_{2} & = & 1 . \end{matrix}

(1)

It is easy to see that this system has four solutions. Thus, there are four Puiseux series in t:

x = x_{i} (t), i = 1, \dots, 4

converging to (0,0) as

t \to 0

.

Thus, there is a one-to-one correspondence between the multiplicity and the number of Puiseux series expansions of the zero set of the perturbed system near

z_{0}

.

Remark 1.

Write

P_{1} = C + H

, where C is the vector containing all of the constant terms of P, and

H = P_{1} - C

. When we derive a system of equations for solving the

(x_{10}, \dots, x_{n 0})

and the αs as in (1), the terms in H are all higher order terms and will not contribute anything. So, in the perturbation, only the constant terms are needed.

In general, we have the following:

Proposition 1.

Suppose that

z_{0} \in C^{s}

is an isolated zero of P. Then, there is a one-to-one correspondence between the multiplicity and the number of Puiseux series expansions of the zero set of the perturbed system

P + t C

near

z_{0}

, where

C \in C^{s}

is a generic vector, which is the constant term of a generic polynomial

P_{1}

.

Proof.

Assume that

z_{0}

is a zero of P with multiplicity m. Let

x_{i} (t) = c_{i 0} t^{α} + c_{i 1} t^{2 α} + h i g h e r o r d e r t e r m s

,

i = 1, \dots, k

and

y_{i} (t) = b_{i 0} t^{α} + b_{i 1} t^{2 α} + h i g h e r o r d e r t e r m s

, and

i = 1, \dots, K

be the Puiseux series expansion of the solution set of

P (x) + t C = 0

and

P (x) + t P_{1} = 0

, respectively, near

t = 0

, where

c_{i 0} \neq c_{j 0}

and

b_{i 0} \neq b_{j 0}

. According to Remark 1,

k = K

and

b_{i 0} = c_{i 0}

. Since for each

t > 0

small enough, the system

P (x) + t P_{1} = 0

has exactly m distinct solutions, and

y_{i} (t)

’s are distinct solutions of the system, where

m = k = K

. □

In [1], Stetter described the following:

Definition 1

(Definition 9.13, [1]). Suppose that

P_{0}

is a singular system of s polynomials in s variables with a d-dimensional zero manifold

M_{0}

. For a specified near singular polynomial system,

P_{t} (z) : = P_{0} (z) + t P_{1} (z),

let

z = z (t)

be such that

P_{t} (z (t)) \equiv 0

. If

z (t) = z_{0} + z_{1} t + \dots

,

r a n k P_{0}^{'} (z_{0}) = s - d

, and

P_{1} (z_{0}) \in r a n g e P_{0}^{'} (z_{0})

, then

z_{0} \in M_{0}

is called a simple distinguished point of

P_{t}

.

Example 3.

Let

P_{0} (x, y) = (\begin{matrix} x \\ x y \end{matrix}), P_{1} (x, y) = - (\begin{matrix} c_{1} \\ c_{2} \end{matrix}),

and

M_{0} = {(x, y) | x = 0}

. Let

z_{0} = (0, c_{2} / c_{1})

and

z (t) = z_{0} + (c_{1} t, 0)

. Then,

P_{t} (z (t)) \equiv 0

, and

P_{1} (z_{0}) = P_{0}^{'} (z_{0}) {(- c_{1}, - c_{1} / c_{2})}^{t}

. So,

P_{1} (z_{0}) \in r a n g e P_{0}^{'} (z_{0})

, and thus,

z_{0}

is a simple distinguished point of

P_{t}

.

Note that the main characteristic of these kinds of points is that they depend on

P_{1}

. The following example shows, however, that these kinds of points do not have to be “simple”.

Example 4.

Let

P_{0} (x, y, z) = (\begin{matrix} x^{2} \\ y^{2} \\ x y z \end{matrix}), P_{1} (x, y, z) = - (\begin{matrix} c_{1} \\ c_{2} \\ c_{3} \end{matrix}) .

Then,

M_{0} = {(x, y, z) | x = y = 0}

. It is not too hard to see that there are four curves

(x_{i} (t), y_{j} (t), z_{i j} (t))

, of

P_{t} (x (t), y (t), z (t)) = 0

, where

x_{i} (t) = {(- 1)}^{i} \sqrt{c_{1} t}, i = 1, 2

,

y_{j} (t) = {(- 1)}^{j} \sqrt{c_{2} t}, j = 1, 2

, and

z_{i j} (t) = {(- 1)}^{i + j} c_{3} / \sqrt{c_{1} c_{2}}

, thus converging to the points

z_{0 k} = (0, 0, {(- 1)}^{k} c_{3} / \sqrt{c_{1} c_{2}}), k = 1, 2

. This means for small t, the system

P_{0} (x) + t P_{1} (x) = 0

has two solutions near

z_{0 k}, k = 1, 2

. Also note that

z_{0}

depends on the perturbation

P_{1}

. In this case,

s = 3, d = 1

. While

r a n k (P_{0}^{'} (z)) = 0 < 3 - 1

for all

z \in M

, only two points are really “distinguished” and are dependent on the perturbation.

Motivated by the examples such as the one above, we introduce a new definition below:

Definition 2.

A generalized distinguished point

z_{0} \in M_{0}

with respect to

P_{0}

under the perturbation

P_{1}

is such that there are m (

\geq 1

) solution curves of

P_{0} (x) + t P_{1} (x) = 0

converging to

z_{0}

as

t \to 0

. Also,

z_{0}

depends on the perturbation

P_{1}

. The multiplicity of such points

z_{0}

with respect to this perturbation is defined as m.

In [1], Stetter also described the following:

Definition 3

(Definition 9.14, [1]). Suppose that

P_{0}

is a singular system of s polynomials in s variables with a d-dimensional zero manifold

M_{0}

. Points of

z \in M_{0}

with

r a n k (P_{0}^{'} (z)) < s - d

are multiple distinguished points of

P_{0}

.

Definition 4

(Definition 9.15, [1]). Suppose that

P_{0}

is a singular system of s polynomials in s variables with a d-dimensional zero manifold

M_{0}

, and

z \in M_{0}

is a multiple distinguished point of

P_{0}

. For the sake of simplicity, we assume that all the zeros of

P_{0}

are contained in

M_{0}

, and z is the only distinguished point of

P_{0}

, multiple or not. Let

I_{0}

be the ideal of polynomials that vanish on

M_{0}

,

N_{0}

is a normal set (see Appendix A) of

I_{0}

, and N is a normal set of

< P_{0} >

containing

N_{0}

. Then, z has multiplicity

# (N \ N_{0}) + 1

.

Example 5.

Let

P_{0} (x, y) = (\begin{matrix} x^{2} \\ x y^{2} \end{matrix}) .

Then,

M_{0} = {(x, y) | x = 0}

,

z_{0} = (0, 0)

,

I_{0} = < x >

, and a normal set of

I_{0}

and

N_{0}

is

{1, y, y^{2}, y^{3}, \dots}

. In this case,

s = 2, d = 1

, and

r a n k P^{'} (z_{0}) = 0 < 2 - 1

. So,

z_{0}

is the multiple distinguished point of

P_{0}

. However, a normal set N of

< P_{0} >

is

{1, x, x y, y, y^{2}, y^{3}, \dots}

. Thus,

N \ N_{0} = {x, x y}

. So, according to Definition 4, the multiplicity is suppose to be

2 + 1 = 3

.

Now, let

P_{1} (x, y) = - (\begin{matrix} c_{1} \\ c_{2} \end{matrix})

and

Q_{t} = P_{0} + t P_{1}

. It is not too hard to see that there are four curves

(x_{i} (t), y_{i j} (t))

such that

Q_{t} (x (t), y (t)) \equiv 0

, where

x_{i} (t) = {(- 1)}^{i} \sqrt{c_{1} t}, i = 1, 2

,

y_{i j} (t) = {(- 1)}^{j} \sqrt{{(- 1)}^{i} c_{2} \sqrt{t / c_{1}}}, j = 1, 2

. It is also easy to see that they all converge to the point

z_{0} = (0, 0)

. Thus, the multiplicity of

z_{0}

should be four instead of three.

Based on the observations from examples such as the one above, we will modify the definition of distinguished points and give a new approach to calculate the multiplicity.

Definition 5.

Let

P_{0} (x)

be a system of s polynomials with

x \in C^{s}

, and M is a zero manifold of

P_{0}

. If

z_{0} \in M

is such that for a generic perturbation

P_{1}

, the system

P_{0} (x) + t P_{1} (x) = 0

has

m \geq 1

solutions near

z_{0}

for small t, then we call

z_{0}

a proper distinguished point of

P_{0}

, and m is its multiplicity.

Example 6.

Let

P (x, y, z) = (\begin{matrix} x^{2} \\ y^{4} \\ x y z \end{matrix}), P_{1} (x, y, z) = - (\begin{matrix} c_{1} \\ c_{2} \\ c_{3} \end{matrix}) .

Then,

M = {(x, y, z) | x = y = 0}

. It is not too hard to see that there are eight curves

(x_{i} (t), y_{j} (t), z_{i j} (t))

, where

x_{i} (t) = {(- 1)}^{i} \sqrt[2]{c_{1} t}, i = 1, 2

,

y_{j} (t) = u^{j} \sqrt[4]{c_{2} t}, j = 1, 2, 3, 4

, and

z_{i j} (t) = c_{3} t / (x_{i} (t) * y_{j}^{2} (t))

, where u is a fourth primitive root of unity converging to the proper distinguished point

z_{0} = (0, 0, 0)

. This means for small t, the system

P (x) + t P_{1} (x) = 0

has eight solutions near

z_{0}

. In this case,

s = 3, d = 1

. While

r a n k (P_{0}^{'} (z)) = 0 < 3 - 1

for all

z \in M

, only one point is really “distinguished” and is independent of the perturbation. So, the multiplicity of

z_{0}

is eight.

Remark 2.

Suppose that

z_{0} \in M

is a proper distinguished point of

P_{0}

; then, if we expand the zero set of

P_{0} (x) + t P_{1}

near

z_{0}

as Puiseux series,

x (t) = z_{0} + x_{0} t^{α} + x_{1} t^{2 α} + \dots

, then each component of α is positive, and the number combinations of

x_{0}

and α is the multiplicity.

Example 7.

Let

P_{0} (x, y, z) = (\begin{matrix} x^{2} \\ y^{4} \\ x z + y z^{2} \end{matrix}), P_{1} (x, y, z) = - (\begin{matrix} c_{1} \\ c_{2} \\ c_{3} \end{matrix}) .

Then,

M = {x = y = 0}

. It is not too hard to see that there are eight curves

(x_{i} (t), y_{j} (t), z_{i j} (t))

, with

α = (1 / 2, 1 / 4, 1 / 2)

and eight curves with

α = (1 / 2, 1 / 4, 1 / 4)

converging to the proper distinguished point

z_{0} = (0, 0, 0)

. This means for small t, the system

P_{0} (x) + t P_{1} (x) = 0

has 16 solutions near

z_{0}

. Thus, the multiplicity of

z_{0}

is 16.

On the other hand, if

P_{0} (x, y, z) = (\begin{matrix} x^{2} \\ y^{4} \\ x z + y^{3} z^{2} \end{matrix}),

then, there are only eight curves

(x_{i} (t), y_{j} (t), z_{i j} (t))

, with

α = (1 / 2, 1 / 4, 1 / 2)

converging to the proper distinguished point

z_{0} = (0, 0, 0)

. So, the multiplicity is only eight.

Finally, why do some of the point(s) in M become “distinguished”, while the others are not? Can we find it out without making use of something “external”, such as Puiseux series? We need the following:

Definition 6.

Let I and J be two ideals. Define

J : I = {f | f i \in J, \forall i \in I}

Lemma 1

(Proposition 10, Section 4.4 and Lemma 8, Section 4.7, [5]). Let I,

I_{i}

, J, and

J_{i}

be ideals. Then,

\begin{matrix} (⋂_{i = 1}^{r} I_{i}) : J = ⋂_{i = 1}^{r} (I_{i} : J) . \end{matrix}

If Q is primary and

\sqrt{Q} = P

, then

\begin{matrix} i f f \in Q, t h e n Q : < f > = < 1 >, \\ i f f \notin Q, t h e n Q : < f > i s P - p r i m a r y, \\ i f f \notin P, t h e n Q : < f > = Q . \end{matrix}

Proposition 2.

Let

< P > = < p_{1}, \dots, p_{s} >

, and

< P_{c} > = < p_{2} - c_{2} p_{1}, p_{3} - c_{3} p_{1}, \dots, p_{s} - c_{s} p_{1} >

. Let

T_{1} = < P_{c} > : < P >

, and

T_{i + 1} = T_{i} : < P >

. Then,

T_{i + 1} = T_{i} : < p_{1} >

, and there exists an n so that

T_{n + 1} = T_{n}

and

V (T_{n}) = \bar{V (P_{c}) - V (P)}

.

Proof.

Since

T_{i + 1} \supset T_{i}

for all i, then by the ascending chain condition, there exists an n so that

T_{n + 1} = T_{n}

. Since

< P > = < P_{c} > + < p_{1} >

. Then,

T_{1} = < P_{c} > : < P > = (P_{c} : P_{c}) ⋂ (< P_{c} > : < p_{1} >) = < P_{c} > : < p_{1} >

. Suppose

T_{i} = T_{i - 1} : < p_{1} >

. Since

T_{i} \supset < P_{c} >

,

T_{i + 1} = T_{i} : < P_{c} > ⋂ T_{i} : < p_{1} > = T_{i} : < p_{1} >

. Let

< P_{c} > = ⋂_{i = 1}^{m} Q_{i}

be a minimal primary decomposition for

< P_{c} >

. Let

S_{1} = {i | p_{1} \in Q_{i}}

,

S_{2} = {i | p_{1} \notin Q_{i}, p_{1} \in \sqrt{Q_{i}}}

, and let

S_{3} = {i | p_{1} \notin \sqrt{Q_{i}}}

. By Lemma 1,

\forall i \in S_{1}

,

Q_{i} : p_{1} = < 1 >

and

\forall i \in S_{3}

,

Q_{i} : p_{1} = Q_{i}

. Thus,

T_{1} = (⋂_{i \in S_{3}} Q_{i}) ⋂ (⋂_{i \in S_{2}} (Q_{i} : < p_{1} >)

. It follows from exercise 4.10 of Chapter 4 in [5] for each

i \in S_{2}

,

Q_{i} : < p_{1}^{n} > = < 1 >

. Thus,

T_{n} = ⋂_{i \in S_{3}} Q_{i}

. Now, let

R_{0} = ⋂_{i = 1}^{m} \sqrt{Q_{i}}

, and define

R_{i} = R_{i - 1} : < p_{1} >

. Since

R_{0}

is redical,

R_{1} = R_{2} = \dots = R_{n}

. On the other hand,

R_{1} = ⋂_{i \in S_{3}} \sqrt{Q_{i}}

. It follows from Theorem 7 of Section 4.4 in [5] that

V (R_{n}) = \bar{V (R_{0}) - V (P)} = ⋃_{i \in S_{3}} V (\sqrt{Q_{i}}) = V (T_{n})

, since

V (R_{0}) = V (P_{c})

,

V (< P_{c} > : < p_{1}^{n} >) = \bar{V (P_{c}) - V (P)}

. □

Example 8.

In Example 6,

Q_{t} = (x^{2} - c_{1} t, y^{4} - c_{2} t, x y z - c_{3} t)

. Now, let us look at the subideal of

< P_{0} >

generated by eliminating the variable t from the polynomials in

Q_{t}

:

< Q_{0} (c) > = < y^{4} - d_{1} x^{2}, x y z - d_{2} x^{2} >, d_{1} = c_{2} / c_{1}, d_{2} = c_{3} / c_{1} .

It is not too hard to see that for generic c values,

Q_{0} (c)

’s zero set contains eight branches of curves not contained in M, which are the projections of the curves in Example 6, thus passing through

z_{0} = (0, 0, 0) \in M

. To “filter” out the curves, let

Q = Q_{0} : < x^{2} >

.

Q = < y^{4} - d_{1} x^{2}, - y z + d_{2} x, - z^{2} * d_{1} + y^{2} d_{2}^{2}, - d_{1} x z + d_{2} y^{3} >

. Let

R = < P_{0} > + Q

. Then, the multiplicity of

z_{0}

with respect to R is eight.

Example 9.

In Example 4,

Q_{t} = (x^{2} - c_{1} t, y^{2} - c_{2} t, x y z - c_{3} t)

. Let

< Q_{0} (c) > = < y^{4} - d_{1} x^{2}, x y z - d_{2} x^{2} >, d_{1} = c_{2} / c_{1}, d_{2} = c_{3} / c_{1} .

Let

Q = Q_{0} : < x^{2} >

. Then,

Q = < - y^{2} + d_{1} x^{2}, - y d_{2} + x z d 1, y z - d_{2} x >

. The intersection of the zero set of Q and M contains two points

z_{0 k} = (0, 0, {(- 1)}^{k} c_{3} / \sqrt{c_{1} c_{2}}), k = 1, 2

. Let

R = < P_{0} > + Q

. Then, the multiplicity of

z_{0 k}

with respect to R is four for

k = 1, 2

.

In general, we have the following.

Theorem 1.

Let P be a system of s polynomials with s variables. Suppose that M is a zero manifold of P. Let c be a generic vector, and let

Q_{0} (c) = {(q_{2}, \dots, q_{s})}^{t}

, where

q_{i} = p_{i} - d_{i} p_{1}, i = 2, \dots, s

,

d_{i} = c_{i} / c_{1}

,

Q_{1} = Q_{0} : < p_{1} >

, …,

Q_{i} = Q_{i - 1} : < p_{1} >

. Then, there exists an n so that

Q_{n + 1} = Q_{n}

. Denote

Q_{n} = Q

. Suppose that

z_{0} \in M

is a zero of

R = < P > + Q

. Then,

z_{0}

is an isolated zero of

R = < P > + Q

, and the multiplicity of

z_{0}

as a zero of R is the same as the multiplicity of

z_{0}

as a distinguished point of P. In addition, if

z_{0}

is independent of c, then

z_{0}

is a proper distinguished point of P; otherwise, it is a generalized distinguished point of P.

Proof.

According to Proposition 2,

z_{0} \in \bar{V (P_{c}) - V (P)} ⋂ V (P)

. However, the only points in

\bar{V (P_{c}) - V (P)} ⋂ V (P)

are the end points

z (0)

of the curves

z (t)

, where

P (z (t)) = c t

. There are only finitely many of them, so they are isolated. Suppose that the multiplicity of

z_{0}

as an isolated zero of R is m. Let

P_{t} = (p_{1} - c_{1} t, p_{2} - c_{2} t, \dots, p_{s} - c_{s} t)

and

R_{t} = < P_{t} > + Q

. For each t,

V (P_{t}) = V (Q_{t}) \subset V (Q)

consists of isolated points. Since the multiplicity of

z_{0}

as a zero of

R_{0}

is m, for small t, there are m points of the zero set of

P_{t}

near

z_{0}

. Thus, by definition, the multiplicity of

z_{0}

as a distinguished point of P is m. □

Example 10.

Let

Q_{t} (x, y, z) = (x^{2} - c_{1} t, y^{4} - c_{2} t, x z + y z^{2} - c_{3} t)

, where

(x, y, z) \in C^{3}

. Let

< Q_{0} (c) > = < y^{4} - c_{2} x^{2} / c_{1}, x y z - c_{3} x^{2} / c_{1} >

and

Q = Q_{0} : < x^{2} >

. Then,

Q = < y^{4} - d_{2} x^{2}, - x z - y z^{2} + d_{3} x^{2}, - x z^{2} d_{2} - y^{3} z + x y^{3} d_{3}, z y^{2} - y^{2} x d_{3} + z y^{3} d_{3} - z^{3} d_{2} >,

where

d_{i} = c_{i} / c_{1}

. The multiplicity of

z_{0} = (0, 0, 0)

as an isolated zero of

R = < P_{0} > + Q_{1}

is 16, and so is the multiplicity of

z_{0}

as a distinguished point of

P_{0}

.

Author Contributions

Conceptualization, X.W.; methodology, X.W.; software, X.W.; validation, X.W.; formal analysis, X.W.; investigation, X.W.; resources, X.W.; data curation, X.W.; writing—original draft preparation, X.W.; writing—review and editing, W.L.; visualization, X.W.; supervision, X.W.; project administration, X.W. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Conflicts of Interest

The authors declare no conflict of interest.

Appendix A

Definition A1.

A set of monomials

N = {x^{j} | j \in J}

, where

x \in C^{s}

and

j = (j_{1}, \dots, j_{s})

, is called closed if

x^{j} \in N \Rightarrow x^{j^{'}} \in N \forall j^{'} : x^{j^{'}} | x^{j} .

Definition A2.

Let I be a polynomial ideal and

R (I)

be the quotient ring. A closed set of monomials

N (I)

is a normal set of I if the monomials in

N (I)

form a basis of

R (I)

.

References

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A Note on the Multiplicity of the Distinguished Points

Abstract

Author Contributions

Funding

Conflicts of Interest

Appendix A

References

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