Abstract
AbstractLet $$\Sigma (f)$$
Σ
(
f
)
be the singular points of a polynomial $$f \in \mathbb {K}[x,y]$$
f
∈
K
[
x
,
y
]
in the plane $$\mathbb {K}^2$$
K
2
, where $$\mathbb {K}$$
K
is $$\mathbb {R}$$
R
or $$\mathbb {C}$$
C
. Our goal is to study the singular point map $$\mathfrak {S}_d$$
S
d
, it sends polynomials f of degree d to their singular points $$\Sigma (f)$$
Σ
(
f
)
. Very roughly speaking, a polynomial f is essentially determined when any other g sharing the singular points of f satisfies that $$f = \lambda g$$
f
=
λ
g
; here both are polynomials of degree d, $$\lambda \in \mathbb {K}^* $$
λ
∈
K
∗
. In order to describe the degree d essentially determined polynomials, a computation of the required number of isolated singular points $$\delta (d)$$
δ
(
d
)
is provided. A dichotomy appears for the values of $$\delta (d)$$
δ
(
d
)
; depending on a certain parity, the space of essentially determined polynomials is an open or closed Zariski set. We compute the map $$\mathfrak {S}_{3}$$
S
3
, describing under what conditions a configuration of 4 points leads to a degree 3 essentially determined polynomial. Furthermore, we describe explicitly configurations supporting degree 3 non essential determined polynomials. The quotient space of essentially determined polynomials of degree 3 up to the action of the affine group $$\hbox { Aff}\hspace{1.42271pt}(\mathbb {K}^2)$$
Aff
(
K
2
)
determines a singular $$\mathbb {K}$$
K
-analytic surface.
Publisher
Springer Science and Business Media LLC
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