International Journal of Mathematics and Mathematical Sciences Volume 2010, Article ID 962719,22pages

doi:10.1155/2010/962719

*Research Article*

**On the Complex and Real Hessian Polynomials**

**Emigdio Mart´ınez-Ojeda**

**1**

_{and Adriana Ortiz-Rodr´ıguez}

_{and Adriana Ortiz-Rodr´ıguez}

**2**

*1 _{Universidad Aut´onoma de la Ciudad de M´exico (UACM), Calzada Ermita Iztapalapa 4163,}*

*Col. Lomas de Zaragoza C.P., Delegaci´on Iztapalapa, 09620 M´exico, DF, Mexico*

*2 _{Instituto de Matem´aticas (IMATE-D.F.), Universidad Nacional Aut´onoma de M´exico (UNAM),}*

*Area de la Inv. Cient. Circuito Exterior, C.U., C.P, 04510 M´exico, DF, Mexico*

Correspondence should be addressed to Adriana Ortiz-Rodr´ıguez,aortiz@matem.unam.mx

Received 25 November 2009; Accepted 8 March 2010

Academic Editor: Mihai Putinar

Copyrightq2010 E. Mart´ınez-Ojeda and A. Ortiz-Rodr´ıguez. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study some realization problems related to the Hessian polynomials. In particular, we solve the Hessian curve realization problem for degrees zero, one, two, and three and the Hessian polynomial realization problem for degrees zero, one, and two.

**1. Introduction**

A topic which has been of interest since the XIX century is the study of the parabolic curve of smooth surfaces in real three-dimensional space, as shown in the works of Gauss, Darboux, Salmon1, Kergosien and Thom2, Arnold3, among others.

The parabolic curve of the graph of a smooth function, *f* : R2 _{→} _{R}_{, is the set}
{*p, fp* ∈ R2_{×}_{R}_{: Hess}_{f}_{}_{p}_{0}_{}}_{, where Hess}_{f}_{:}_{} _{f}

*xxfyy*−*fxy*2 *. In this case, the Hessian*

*curve of* *f*, Hess*fx, y* 0, is a plane curve which is the projection of the parabolic curve
into the*xy*-plane along the*z*-axis. When*f*is a polynomial of degree*n*in two variables, Hess

*f*is a polynomial of degree at most 2*n*−4. Therefore, the Hessian curve is an algebraic plane
curve. In this setting there are two natural realization problems related to the Hessian of a
polynomial.

1*Hessian curve realization problem. Given an algebraic plane curvegx, y* 0 inK2_{,}
whereKR or C, we ask: When is*gx, y* 0 the Hessian curve of a polynomial

*f* ∈K*x, y*?

2*Hessian polynomial realization problem. Given* *g* ∈ K*x, y*we ask: When does *f* ∈

K*x, y*exist such that Hess*f* *g*? If such *f* exists andK C K R, then

Note that problem 2 contains problem 1. Moreover, in the real case, problem 2 is a global realization problem of a smooth function such as the Gaussian curvature function. In

5, Arnold studies this problem locally.

This work is devoted to problems 1 and 2 for complex and real case of degree less or
equal to three. It is divided in two parts. In the first we give the results according to the degree
of the polynomials*g*. In the second part, we give the proofs and some other results such as a
geometric interpretation of Corollary2.8.

**2. Notation and Results**

For each nonnegative integer number*n,*we define AK* _{n}* :{

*f*∈K

*x, y*| deg

*f*≤

*n*}. Let us

*consider the Hessian map*

*H _{n}*K:AK

*−→ AK*

_{n}_{2}

_{n}_{−}

_{4}

*,*given by

*f*−→Hess

*f.*2.1

We will say that the Hessian map *H _{n}*K

*is complex or real if*K C or K R, respectively. We remark that dimAK

*≥dimAK*

_{n}_{2}

_{n}_{−}

_{4}if 2 ≤

*n*≤ 4 and dimAK

*dimAK*

_{n}<_{2}

_{n}_{−}

_{4}if

*n*≥5. In particular, for

*n*≥4, the image,

*H*KAK

_{n}*is a connected subset of codimension at least three inAK*

_{n},_{2}

_{n}_{−}

_{4}. This means that, in “general”, a polynomial

*g*∈AK

_{2}

_{n}_{−}

_{4}is not a Hessian polynomial if

*n*≥4 under

*H*K.

_{n}In diagram form we have

AK

2 AK3 AK4 · · · AK*n*· · ·
↓*H*_{2}K↓*H*_{3}K↓*H*_{4}K· · · ↓*Hn*K· · ·

AK

0 AK2 AK4 · · · AK2*n*−4· · ·

2.2

*In virtue of the previous remarks, we introduce the fiber of* *g* ∈AK_{2}_{n}_{−}_{4}*underH _{n}*Kas the
set

*H _{n}*K−1

*g*:

*f*∈AK

*|*

_{n}*H*K

_{n}*fg.*2.3

Even when we consider the fibers, *H _{n}*K−1

*g*, for diﬀerent values of

*n*, we are interested in knowing if the fiber

*H*K−1

_{n}*g*is not empty when

*n*satisfies 0 ≤ 2

*n*−4−deg

*g*≤ 1

*.*

If the fiber *H _{n}*K−1

*g*is empty, the next problem is to see if the fibers

*H*K−1

_{s}*g*are not empty for

*s*≥

*n*1 this problem will not be studied in this work. Another way to study the Hessian polynomial realization problem is by describing the set of all polynomials

*f*∈AK

*, with*

_{r}*r*≥

*n*

*m*4

*/*2

*,*such that

*H*K

_{n}*f*

*g*whenever

*g*∈AK

*.*

_{m}*Remark 2.1. Letgx, y* 2_{i}_{}*n _{j}*−

_{}4

_{0}

*bijxiyj*be a polynomial in AK

_{2}

_{n}_{−}

_{4}. There exists a polynomial

*fx, y* *n _{i}*

_{}

_{j}_{}

_{0}

*aijxiyj*in

*Hn*K−1

*g*if and only if

*f*satisfies the following system of2

*n*−

3*n*−1equations:

where*hijars*are the coeﬃcients of the Hessian polynomial *Hn*K*f*. They are also quadratic

polynomials in the variables *ars*.

It is important to note that the computations for solving the system2.4are generally very complicated.

In the following results we describe the sets of polynomials of a given degree which are Hessian and those which are not Hessian under a specific Hessian map.

**2.1. Degree of**

**2.1. Degree of**

*g*

**Equal to Zero**

**Equal to Zero**

* Proposition 2.2. For eachg*∈AC

_{0}

*, the setH*

_{2}C−1

*gis a quadric in*AC

_{2}

*given by*

*a*2

114*a*20*a*02−*g.*
*Moreover, this quadric is singular if and only ifg0.*

**Corollary 2.3. In the complex case every element in***A*C_{0} C *is a complex Hessian polynomial*
*underH*_{2}C*. And, in the real case every element inA*R_{0} R*is a real Hessian polynomial underH*_{2}R*.*

* Proposition 2.4. For eachg*∈AC

_{0}

*, the setH*

_{3}C−1

*gis an analytic subvariety in*AC

_{3}

*which is given*

*by the union of connected analytic subvarieties which are parametrized by the following.*

1*F*_{1}± :C×C∗×C∗×C3 _{→} _{A}C

3;*t*1*, t*2*, t*3*, t*4*, t*5*, t*6→*a*30*t*22*/*3*t*3*, a*21*t*2*, a*12*t*3*, a*03

*t*2

3*/*3*t*2*, a*20 *t*1*, a*11 2*t*3*t*1±*t*2√−*g/t*2*, a*02 *t*3*t*3*t*1±*t*2√−*g/t*22*, a*10 *t*4*, a*01

*t*5*, a*00*t*6*.*

2*F*2 : C×C∗×C3 → AC_{3}*;t*1*, t*2*, t*3*, t*4*, t*5 → *a*30 0*, a*21 0*, a*12 0*, a*03 0*, a*20

*t*2

1*g/*4*t*2*, a*11 *t*1*, a*02*t*2*, a*10*t*3*, a*01 *t*4*, a*00*t*5*.*

3*F*_{3}± : C5 _{→} _{A}C

3;*t*1*, t*2*, t*3*, t*4*, t*5 → *a*30 *t*1*, a*21 0*, a*12 0*, a*03 0*, a*20 *t*2*, a*11
±√−*g, a*02 0*, a*10 *t*3*, a*01*t*4*, a*00*t*5*.*

4*F*_{4}± : C5 _{→} _{A}C

3*;t*1*, t*2*, t*3*, t*4*, t*5 → *a*30 0*, a*21 0*, a*12 0*, a*03 *t*1*, a*20 0*, a*11
±√−*g, a*02 *t*2*, a*10*t*3*, a*01 *t*4*, a*00 *t*5*.*

**2.2. Degree of**

**2.2. Degree of**

*g*

**Equal to One**

**Equal to One**

**Proposition 2.5. For each**gx, y*b*10*xb*01*yb*00 ∈AC_{2} *of degree one, the set* *H*_{3}C−1*g* *is*
*an analytic subvariety in*AC_{3} *which is given by the union of analytic subvarieties parametrized by the*
*following.*

1*For* *b*10*b*01 *0,* *F*1 : C∗×C×C3 \ {4*b*201*t*12 −4*b*10*b*01*t*1*t*2 *b*210*t*22 *b*00*b*210 0} →

AC

3*;t*1*, t*2*, t*3*, t*4*, t*5→*a*30 1*/*3*b*310*t*1*/*4*b*201*t*21−4*b*10*b*01*t*1*t*2*b*210*t*22*b*00*b*210*, a*21

*b*01*b*2_{10}*t*1*/*4*b*2_{01}*t*2_{1}−4*b*10*b*01*t*1*t*2*b*2_{10}*t*2_{2}*b*00*b*2_{10}*, a*12 *b*2_{01}*b*10*t*1*/*4*b*2_{01}*t*2_{1}−4*b*10*b*01*t*1*t*2

*b*2

10*t*22*b*00*b*210*, a*03*b*013 *t*1*/*34*b*201*t*21−4*b*10*b*01*t*1*t*2*b*210*t*22*b*00*b*210*, a*20*t*1*, a*11*t*2*, a*02

*t*2_{2}*b*00*/*4*t*1*, a*10*t*3*, a*01 *t*4*, a*00*t*5*.*

2*For* *b*10*b*01*/0,* *F*2± : C∗ ×C3 → AC3*;t*1*, t*2*, t*3*, t*4 → *a*30 1*/*3*t*1*b*10*/b*01*, a*21

*t*1*, a*12 *b*01*t*1*/b*10*, a*03 1*/*3*b*2_{01}*t*1*/b*_{10}2 *, a*20 0*, a*11 ± −*b*00*, a*02

1*/*4*b*01±4*t*1 −*b*00*b*10*/b*10*t*1*, a*10 *t*2*, a*01*t*3*, a*00*t*4*.*

From this proposition we have the following.

**2.3. Degree of**

**2.3. Degree of**

*g*

**Equal to Two**

**Equal to Two**

Let *g*1*, g*2 ∈ K*x, y*. We say that *g*1 *is in the orbit of* *g*2 if they are equivalent by an aﬃne
transformation of the planeK2_{.}

**Theorem 2.7.**

*Complex Case*

1*The complex polynomials of degree two*

*y*2 _{x}_{−}* _{y}*2

*2*

_{y}_{−}

*2*

_{x}_{−}

_{k, k}_{∈}

_{C}

_{}

_{2.5}

_{}

*are complex Hessian polynomials under* *H*_{3}C*. Moreover, a polynomialg*∈AC_{2} *of degree two*
*is a complex Hessian polynomial under* *H*_{3}C*if and only if it belongs to the orbit of one of*
*those polynomials.*

2*The complex polynomials,* *y*2 *r* ∈ AC_{2}*,* *where* *r* ∈ C∗*,* *are not complex Hessian*
*polynomials under* *H*_{3}C*. Moreover, a polynomialg* ∈ AC_{2} *of degree two is not a complex*
*Hessian polynomial under* *H*_{3}C *if and only if it belongs to the orbit of one of those*
*polynomials.*

*Real case*

1*The real polynomials of degree two*

−*y*2 _{x}_{−}* _{y}*2

*2*

_{y}_{−}

*2*

_{x}_{−}

_{r}1*, r*1*>*0 −*y*2−*x*2*r*2*, r*2*>*0 2.6

*are real Hessian polynomials underH*_{3}R*. Moreover, a polynomialg* ∈ AR_{2} *of degree two is*
*a real Hessian polynomial underH*_{3}R*if and only if it belongs to the orbit of one of those*
*polynomials.*

2*The real polynomials of degree two*

*y*2 * _{y}*2

_{−}

*2*

_{x}_{−}

_{r}3*, r*3≤0 *y*2*x*2−*r*4*, r*4∈R −*y*2−*x*2*r*5*, r*5≤0

*y*2_{}_{x}* _{y}*2

_{−}

_{r}6*, r*6 ∈R∗ −*y*2−*r*7*, r*7∈R∗

2.7

*are not real Hessian polynomials under* *H*_{3}R*. Moreover, a polynomialg*∈AR_{2} *of degree two*
*is not a real Hessian polynomial under* *H*_{3}R*if and only if it belongs to the orbit of one of*
*those polynomials.*

It is well known that the complex aﬃne classification of conics is given by the normal
forms:*y*2 _{} _{x}_{}_{parabola}_{}_{,}* _{y}*2

_{}

*2*

_{x}_{}

_{1}

_{}

_{general conic}

_{}

_{,}

*2*

_{y}_{}

*2*

_{x}_{}

_{line pair}

_{}

_{,}

*2*

_{y}_{}

_{1}

_{}

_{parallel}lines, and finally

*y*20double line. Therefore, we have the following corollary.

**Corollary 2.8. All the complex a**ﬃne conics, except the parallel lines, are complex Hessian curves of*polynomials in*AC_{3}*.*

**Proposition 2.9. Let***gx, y* *b*20*x*2 *b*11*xy* *b*02*y*2 *b*00 *be a polynomial in* AC_{2} *with*

*b*204*b*20*b*02 −*b*2_{11}*/0. Then the setH*_{3}C−1*gis an analytic subvariety on*AC_{3} *which is the union*
*of analytic subvarieties parametrized by the following.*

1*F*_{1}± : C4 _{→} _{A}C

3*;* *t*1*, t*2*, t*3*, t*4 → *a*30 *t*1*, a*21 1*/*23*b*11*t*1±

9*t*2_{1}*b*_{11}2 −36*t*2_{1}*b*02*b*20−*b*3_{20}*/b*20*, a*12*=*−6*t*1*b*02*b*203*b*11*t*1±

9*t*2_{1}*b*2_{11}−36*t*2_{1}*b*02*b*20−*b*3_{20}

*/*2*b*2

20*,* *a*03 *=* −1*/*6*b*220±*b*02

9*t*2

1*b*211−36*t*21*b*02*b*20−*b*203 6*b*11*t*1*b*02 − 3*b*11*t*1 ±

9*t*2

1*b*112 −36*t*21*b*02*b*20−*b*320*b*211*/b*20 3*b*11*b*02*t*1*,* *a*20 *b*20

*b*00*/*4*b*20*b*02−*b*2_{11}*,* *a*11

*b*11

*b*00*/*4*b*20*b*02−*b*2_{11}*,a*02 *b*02

*b*00*/*4*b*20*b*02−*b*2_{11}*,a*10*t*2*,a*01 *t*3*, a*00*t*4*.*

2*F*_{2}±*:* C4 _{→} _{A}C

3*;* *t*1*, t*2*, t*3*, t*4 → *a*30 *t*1*, a*21 3*b*11*t*1 ±

9*t*2_{1}*b*_{11}2 −36*t*_{1}2*b*02*b*20−*b*3_{20}*/*2*b*20*,* *a*12 *=* 3*b*11*t*1 − 6*t*1*b*02*b*20 ±

9*t*2

1*b*112 −36*t*21*b*02*b*20−*b*320*/*2*b*220*,* *a*03 *=* −1*/*6*b*202 ±*b*02

9*t*2

1*b*112 −36*t*21*b*02*b*20−*b*320
3*b*02*b*11*t*1 6*b*11*t*1*b*02 − 3*b*_{11}3*t*1 ± *b*2_{11}

9*t*2

1*b*112 −36*t*21*b*02*b*20−*b*320*/b*20*,* *a*20 *=*
−*b*20

*b*00*/*4*b*20*b*02−*b*2_{11}*,a*11 *=*−*b*11

*b*00*/*4*b*20*b*02−*b*_{11}2 *,a*02 *=*−*b*02

*b*00*/*4*b*20*b*02−*b*2_{11}*,*

*a*10*=t*2*, a*01*=t*3*, a*00*=t*4*.*

From the study of the previous fibers, a natural question arises. What is the relation
between the set of critical points of*H _{n}*Cand the set of singular Hessian curves defined by
polynomials inAC

_{2}

_{n}_{−}

_{4}?

We defineAC* _{nR}*:{

*f*∈C

*x, y*|2≤deg

*f*≤

*n*}. Let us describe the relation between the set of critical points of

*H*

_{3}C

*:AC*

_{R}_{3}

*→ AC*

_{R}_{2}and the set of polynomials inAC

_{2}such that they define singular Hessian curves.

**Proposition 2.10. If***f* ∈ AC_{3}_{R}*is a critical point of the mapH*_{3}C* _{R}* : AC

_{3}

*→ AC*

_{R}_{2}

*, then the Hessian*

*curveH*

_{3}C

_{R}fx, y*0 is singular or it has degree one.*

For the general case, we have the following conjecture.

**Conjecture 2.11. If***f* ∈ A*nRis a critical point of the mapH _{nR}*C : AC

*→ AC*

_{nR}_{2}

_{n}_{−}

_{4}

*, then its Hessian*

*curve, Hessfx, y* 0*,is singular or has degree less than 2n*−*4.*

**2.4. Degree of**

**2.4. Degree of**

*g*

**Equal to Three**

**Equal to Three**

The following theorem is one of the most important of this paper.

**Theorem 2.12.**

*Complex Case*

1*The curves defined by Table1of complex cubic curves (see [6]) are complex Hessian curves*
*under* *H*_{4}C*.*

2*The curves defined by Table2of complex cubic curves are not complex Hessian curves under*

**Table 1: Complex Hessian cubic curves.**

*xy*2_{}* _{bx}*2

_{}

_{cx}_{}

_{d, b, c, d}_{∈}

_{C}

*2*

_{xy}_{}

_{ey}_{}

*2*

_{bx}_{}

_{cx}_{}

_{d, e}_{∈}

_{C}∗

_{, b, c, d}_{∈}

_{C}

*y*2

_{}

*3*

_{ax}_{}

*2*

_{bx}_{}

_{cx}_{}

_{d, a}_{∈}

_{C}∗

_{, b, c, d}_{∈}

_{C}

**Table 2: Complex cubic curves which are not complex Hessian curves.**

*xy*2_{}_{ey}_{}* _{ax}*3

_{}

*2*

_{bx}_{}

_{cx}_{}

_{d, a}_{∈}

_{C}∗

_{, b, c, d, e}_{∈}

_{C}

_{xy}_{}

*3*

_{ax}_{}

*2*

_{bx}_{}

_{cx}_{}

_{d, a}_{∈}

_{C}∗

_{, b, c, d}_{∈}

_{C}

*yax*3

_{}

*2*

_{bx}_{}

_{cx}_{}

_{d, a}_{∈}

_{C}∗

_{, b, c, d}_{∈}

_{C}

*3*

_{x}_{}

*2*

_{bx}_{}

_{cx}_{}

_{d}_{}

_{0}

_{, b, c, d}_{∈}

_{C}

*Real Case*

1*The normal form of curves (see [7]) defined in Table3is a real Hessian curve under* *H*_{4}R*.*
*Moreover, a curveg* ∈AR_{3} *of degree three is a real Hessian curve under* *H*_{4}R*if and only if*
*its polynomial belongs to the orbit of one of those polynomials.*

2*The normal form of curves defined by the polynomials in Table* *4* *is not a real Hessian*
*polynomial under* *H*_{4}R*. Moreover, a curveg*∈AR_{3} *of degree three is not a real Hessian curve*
*underH*_{4}R*if and only if its polynomial belongs to the orbit of one of those polynomials.*

**3. Proofs**

Let us consider the complex Hessian map*H _{n}*C:AC

*→ AC*

_{n}_{2}

_{n}_{−}

_{4}

*f*→Hess

*f*and recall that the fiber of

*g*∈A2

*n*−4under

*Hn*Cis the set

*Hn*C−1

*g*:{

*f*∈A

*n*C |

*Hn*C

*f*

*g*}

*.*

*Proof of the Proposition2.2. Let* *fx, y* 2_{i}_{}_{j}_{}_{0}*aijxiyj* ∈ AC2. A direct calculus shows that

*fxx*2*a*20,*fxya*11*,*and*fyy*2*a*02. Therefore,

*H*_{2}C*f*4*a*20*a*02−*a*211*.* 3.1

For each*g*∈AC_{0}, consider*Sg* {*a*00*, a*10*, a*01*, a*20*, a*11*, a*02:*a*2_{11}4*a*20*a*02−*g*}. To show that

*H*_{2}C−1*g* *Sg*it is enough to show*H*_{2}C−1*g* ⊂ *S*because a direct substitution shows

*S*⊂ *H*_{2}C−1*g*. Therefore, let us consider*f* ∈*H*_{2}C−1*g*, that is,*H*_{2}C*f* 4*a*20*a*02−*a*2_{11} *g*.
Hence*a*2_{11} 4*a*20*a*02−*g*and the first part of the claim is done. Finally, the derivative of*H*_{2}Cis
given by

*DH*_{2}C*f* 0*,*0*,*0*,*4*a*02*,*−2*a*11*,*4*a*20*.* 3.2

Therefore, we conclude the proof of the result.

*Proof of the Corollary2.3. The polynomialfx, y* *x*2 _{g/}_{4}_{}* _{y}*2

_{satisfies that}

*K*

_{H}**Table 3: Real Hessian polynomials.**

*x*2_{y}_{}* _{y}*2

_{}

_{x}_{}

_{a}1*ya*20*, a*1*, a*2∈R *x*2*yy*2*ya*30*, a*3∈−∞*,*0∪0*,*1*/*4
*x*2_{y}_{}* _{y}*2

_{−}

_{y}_{}

_{a}40*, a*4*<*−3*/*4 *x*2*yy*2−10

*x*2_{y}_{}_{3}_{y}_{}_{0} * _{x}*2

_{y}_{−}

_{3}

_{y}_{}

_{0}

*x*2_{y}_{}_{0} * _{x}*3

_{}

_{a}5*x*−*y*2−10*, a*5∈R*,*
*x*3_{−}* _{y}*2

_{}

_{a}6*x*10*, a*6∈R *x*3−*y*2*x*0

*x*3_{−}* _{y}*2

_{−}

_{x}_{}

_{0}

*3*

_{x}_{−}

*2*

_{y}_{}

_{0}

**Table 4: Real polynomials which are not Hessian polynomials.**

*xy*2_{−}_{x}_{}_{x}_{−}_{3}_{}2_{}_{b}

1*xb*2*y*−*b*30*, b*1*, b*2*, b*3∈R *xy*2−*x*3*b*4*x*2*y*−*b*50*, b*5∈R
*xy*2_{−}* _{x}*3

_{}

_{b}6*x*−10*, b*6∈R *xy*2−*x*3*x*0

*xy*2_{−}* _{x}*3

_{−}

_{x}_{}

_{0}

*2*

_{xy}_{−}

*3*

_{x}_{}

_{0}

*x*3_{}* _{xy}*2

_{−}

_{6}

*2*

_{y}_{}

_{b}7*xb*8*yb*90*, b*7*, b*8*, b*9∈R *x*3*xy*2*b*10*x*3*yb*110*, b*10*, b*11∈R
*x*3_{}* _{xy}*2

_{}

_{b}12*x*10*, b*12∈R *x*3*xy*23*x*0

*x*3_{}* _{xy}*2

_{−}

_{3}

_{x}_{}

_{0}

*3*

_{x}_{}

*2*

_{xy}_{}

_{0}

*x*2_{y}_{}* _{y}*2

_{}

_{y}_{}

_{c}_{}

_{0}

_{, c}_{∈ {0} ∪}

_{}

_{1}

_{/}_{4}

_{,}_{∞}

_{}

*2*

_{x}

_{y}_{}

*2*

_{y}_{−}

_{y}_{}

_{d}_{}

_{0}

_{, d}_{≥ −3}

_{/}_{4}

*x*2_{y}_{}* _{y}*2

_{}

_{1}

_{}

_{0}

*2*

_{x}

_{y}_{}

*2*

_{y}_{}

_{0}

*x*2_{y}_{}_{3}_{x}_{}_{3}_{y}_{}_{b}

130*, b*13∈R *x*2*y*3*y*10

*x*2_{y}_{}_{3}_{x}_{−}_{3}_{y}_{}_{b}

140*, b*14∈R *x*2*y*−3*y*10

*x*2_{y}_{}_{3}_{x}_{}_{1}_{}_{0} * _{x}*2

_{y}_{}

_{3}

_{x}_{}

_{0}

*x*2_{y}_{}_{1}_{}_{0} * _{x}*3

_{−}

_{xy}_{}

_{1}

_{}

_{0}

*x*3_{−}_{xy}_{}_{0} * _{x}*3

_{−}

_{y}_{}

_{0}

*x*3_{}_{b}

15*x*10*, b*15∈R *x*3−3*x*0

*x*3_{}_{3}_{x}_{}_{0} * _{x}*3

_{}

_{0}

* Lemma 3.1. Iffx, y* 3

_{i}_{}

_{j}_{}

_{0}

*aijxiyj*∈AC

_{3}

*, then the mapH*

_{3}C:

*A*C

_{3}→

*A*C

_{2}

*is*

*H*_{3}C*fb*20*x*2*b*11*xyb*02*y*2*b*10*xb*01*yb*00*,* 3.3

*where thebrscoeﬃcients satisfy the following system of quadratic equations:*

*b*20−4*a*22112*a*30*a*12*,*

*b*11−4*a*21*a*1236*a*30*a*03*,*

*b*02−4*a*21212*a*21*a*03*,*

*b*1012*a*30*a*024*a*20*a*12−4*a*21*a*11*,*

*b*0112*a*20*a*034*a*21*a*02−4*a*12*a*11*,*

*b*00−*a*2_{11}4*a*20*a*02*.*

*Proof of the Proposition2.4. Forg* ∈AC_{0} we have that, by Lemma3.1, *H*_{3}C*f* *g*is equivalent
to the following system of equations:

0−4*a*2_{21}12*a*30*a*12*,* 3.5

0−4*a*21*a*1236*a*30*a*03*,* 3.6

0−4*a*2_{12}12*a*21*a*03*,* 3.7

012*a*30*a*024*a*20*a*12−4*a*21*a*11*,* 3.8

012*a*20*a*034*a*21*a*02−4*a*12*a*11*,* 3.9

*g* −*a*2_{11}4*a*20*a*02*.* 3.10

Let*Sg* be the set obtained by the union of the image of parametrizations*F*1*, . . . , F*4. To prove
that*H*_{3}C−1*g* *Sg*, it is enough to show that*H*3C−1*g*⊂*Sg* because a direct substitution

shows that*Sg* ⊂*H*3C−

1_{}_{g}_{}_{. Now, to prove}_{}_{H}_{C}
3−

1_{}_{g}_{}_{⊂}_{S}

*g*we will consider two cases: Case

1is when*a*12*/*0 and Case2is when*a*120.

*Case 1. In this case, from a direct substitution we obtaina*30*a*2_{21}*/*3*a*12,*a*03 *a*2_{12}*/*3*a*21,*a*12 ∈

C∗_{, and}_{}_{3.8}_{}_{,}_{}_{3.9}_{}_{, and}_{}_{3.10}_{}_{. To solve these equations we will assume the following.}

*Subcase 1.1.* *a*20 0. In this case we obtain*a*11±√−*g*;*a*02±*a*12*a*21√−*g/a*2_{21}. All this values
together are contained in the set whose parametrization is*F*1.

*Subcase 1.2.* *a*20*/*0. This case will be subdivided in two subcases.

1*a*02 0. First, we obtain*a*11from3.10. Later, from a substitution of*a*11 together
with the value of*a*30 in 3.8 we get*a*20*a*12 ±*a*21√−*g* 0. This equation implies

*a*112*a*20*a*12±*a*21√−*g/a*21. All these values together are contained in the set whose
parametrization is*F*1.

2*a*02*/*0. In this case, from3.8we obtain

*a*11
1

*a*21*a*12

*a*2_{21}*a*02*a*20*a*212

*.* 3.11

A substitution of*a*11in3.10gives us the quadratic equation in the*a*20variable:

*a*4_{21}*a*2_{02}−2*a*2_{21}*a*2_{12}*a*02

*a*2_{20}*a*4_{12}*ga*2_{21}*a*2_{12}0*.* 3.12

*Case 2. From a direct calculus we obtaina*210 and the equations:

0*a*30*a*03*,*

0*a*30*a*02*,*

0*a*20*a*03*,*

*g* −*a*2_{11}4*a*20*a*02*.*

3.13

To solve these four equations we will consider two cases.

*Subcase 2.1.* *a*02 0. From a direct substitution we get*a*11 ±√−*g*and the two equations:

0*a*30*a*03*,*

0*a*30*a*02*.*

3.14

To solve these two equations we will consider two subcases.

1*a*03 0. We obtain *a*30 ∈ C and *a*20 ∈ C. From all of these values we get the
parametrization*F*3.

2*a*03*/*0. We obtain*a*30*a*200. From all these values we get the parametrization*F*4
when*a*020.

*Subcase 2.2.* *a*02*/*0. We obtain*a*20 *a*2_{11}*g/*4*a*02and the three equations:

0*a*30*a*03*,*

0*a*30*a*02*,*

0*a*20*a*03*.*

3.15

To solve these three equations we will consider two subcases.

1*a*030. We obtain*a*300 and then the parametrization*F*2.

2*a*03*/*0. We obtain*a*30 0,*a*20 0, as well as*a*11 ±√−*g*. All these values together
are included in the parametrization*F*4when*a*02 ∈C∗. Therefore, we have obtained
all parametrizations in the proposition and the proof is done.

Let*g*1*, g*2∈K*x, y*. We say that*g*1*is in the orbit of* *g*2or*g*2*is in the orbit of* *g*1if they
are equivalent by an aﬃne transformation of the planeK2 _{}_{where} _{K}_{}_{R}_{or}_{C}_{}_{.}

*Remark 3.2. If* *g*1 is in the orbit of *g*, then *g*1 is a Hessian polynomial if and only if *g* is a
Hessian polynomial. This remark is due to the equality Hess1*/*det*Tf*◦*T* Hess*f*◦*T,*

*Proof of Corollary2.6. Note that the polynomial* *gx, y* *x* ∈ AC_{2} is a complex Hessian
polynomial under *H*_{3}C because *fx, y* 1*/*12*x*3_{}* _{y}*2

_{∈}

_{}

*C*

_{H}3−

1_{}_{g}_{}_{.}_{On the other hand,}

every complex polynomial of degree one is in the orbit of*g*. By Remark3.2we have that every
complex polynomial of degree one is a Hessian polynomial. The real case is analogous.

*Proof of Proposition2.5. Letgx, y* *b*10*xb*01*yb*00be a polynomial of degree one inAC_{2} with
constant term. Then the expression *H*_{3}C*f* *g*is equivalent, by Lemma3.1, to the system of
equations:

0−*a*2_{21}3*a*30*a*12*,* 3.16

0−*a*21*a*129*a*30*a*03*,* 3.17

0−*a*2_{12}3*a*21*a*03*,* 3.18

*b*1012*a*30*a*024*a*20*a*12−4*a*21*a*11*,* 3.19

*b*0112*a*20*a*034*a*21*a*02−4*a*12*a*11*,* 3.20

*b*00−*a*2114*a*20*a*02*.* 3.21

Denote by*Sg*the union of the images of*F*1and*F*2. We shall prove that *H*_{3}C−1*g*

*Sg.* After some calculus it is proved that *Sg* ⊂ *H*_{3}C−1*g.* Now, we shall prove that

*H*_{3}C−1*g*⊂*Sg.*

Suppose that*a*21*, a*12*/*0. Multiply3.19by *a*21and3.20by *a*12and subtract the two
obtained equations. It gives

4*a*20*a*212−4*a*02*a*22112*a*30*a*02*a*12−12*a*03*a*20*a*21 *b*10*a*12−*b*01*a*21*.* 3.22

From3.16and3.18we obtain *a*2_{21}3*a*30*a*12 and *a*2_{12}9*a*30*a*03*,*respectively, which
we insert in3.22to obtain

*a*12*b*10*b*01*a*21*.* 3.23

From3.23we obtain *a*12 *b*01*a*21*/b*10 if*b*10*/*0 and we insert it in3.16to obtain

*a*30

*b*10*a*21
3*b*01

*.* 3.24

Analogous, we insert it in3.18to obtain

*a*03

*b*2
01*a*21
3*b*2

10

*.* 3.25

*Case 1. Suppose thata*20*/*0. From3.21it has

*a*02

*b*00*a*2_{11}
4*a*20 *.*

3.26

From3.19

*a*12

4*a*21*a*11−12*a*30*a*02*b*10
4*a*20

*.* 3.27

From3.23we obtain *a*21and we replace it in3.27. We replace also *a*30 from3.24and

*a*02to obtain

12*a*2_{20}*b*2_{01}*a*1212*a*11*a*20*a*12*b*10*b*01−3*a*12*b*2_{10}

*b*00*a*2_{11}

3*a*20*b*10*b*2_{01}*.* 3.28

We associate the terms containing *a*12

*a*12

*a*20*b*10*b*2_{01}
4*a*2

20*b*201−4*a*11*a*20*b*10*b*01*b*210*b*00*b*102 *a*211

*.* 3.29

We replace this last expression of*a*12in3.23and we obtain

*a*21

*a*20*b*210*b*01
4*a*2

20*b*201−4*a*11*a*20*b*10*b*01*b*210*b*00*b*102 *a*211

*.* 3.30

We replace the expression of*a*21in3.24and3.25and we have

*a*30

*a*20*b*3_{10}

34*a*2_{20}*b*2_{01}−4*a*11*a*20*b*10*b*01*b*2_{10}*b*00*b*2_{10}*a*2_{11}
*,*

*a*03

*a*20*b*3_{01}
34*a*2

20*b*201−4*a*11*a*20*b*10*b*01*b*210*b*00*b*210*a*211
*.*

3.31

Note that those last four expressions are in the image of *F*1.

*Case 2. Suppose thata*20 0. From3.21we obtain

*a*2_{11}−*b*00*.* 3.32

From3.19

*a*02 4*a*21*a*11*b*10
12*a*30

We replace the value of *a*30obtained from3.24and the expression *a*11in this last equation
to obtain

*a*02

*b*01

±4*a*21 −*b*00*b*10

4*a*21*b*10 *.* 3.34

The expression of *a*12 is obtained from3.23. The expressions of this case are in the image
of *F*2.

*Proof of Theorem2.7.*

*Complex Case*

1Let*gx, y* *ax*2 *bxycy*2*dxeyh* ∈ AC_{2} be a quadratic polynomial. By
an aﬃne transformation of the complex plane the polynomial*g*is equivalent to the
prenormal form *Px, y* *y*2_{−}* _{Ax}*2

_{−}

_{Bx}_{−}

_{C,}_{where}

_{A, B, C}_{∈}

_{C}

_{.}

If*A /*0, then*P*is in the orbit of *y*2_{−}* _{x}*2

_{}

_{k,}_{where}

_{k}_{∈}

_{C}

_{. If}

_{A}_{}

_{0 and}

_{B /}_{}

_{0}

_{,}_{then}

*P* is in the orbit of *y*2_{}_{x}_{. Finally, if} _{A}_{} _{0}_{, B}_{} _{0}_{, C /}_{}_{0}_{,}_{then}_{P}_{is in the orbit of}

*y*2−*m,*where *m*∈C.

Note that the polynomial*fx, y* √−1*/*2*xy*2 _{∈}_{A}C

3 satisfies *H*3C*f* *y*2*.* The
polynomial *fx, y* *xy*2*/*2*x*2*/*2 ∈ AC_{3} verifies *H*_{3}C*f* *x*−*y*2*.* For each

*k* ∈C, the polynomial*fkx, y* *y*3*/*6*x*2*y/*2−
√

−*k/*2*x*2 √_{−}_{k/}_{2}_{}* _{y}*2

_{∈}

_{A}C 3 fulfills

*H*

_{3}C

*fk*

*y*2−

*x*2−

*k.*

2To verify that the polynomials *y*2 _{}_{r,}_{where} _{r}_{∈} _{C}∗_{,}_{are not complex Hessian}

polynomials, we used the computer algebra system Maple 9.5. In particular, the
Groebner package with the graded reverse lexicographic monomial order. We
obtained a reduced Groebner basis for the system *H*_{3}C*f* *y*2_{}_{r .}_{The Groebner}
basis obtained was 1. So, by the Weak Nullstellensatz Theorem there is no solution
to this system of equations.

*Real Case*

Analogous to the complex case, after a composition with an aﬃne transformation of the real
plane, the real quadratic polynomial *gx, y* *ax*2*bxycy*2*dxeyh* is in the orbit of
one of the normal forms: *y*2_{−}* _{x}*2

_{−}

_{q}1*, y*2*x*2− *q*2*,*−*y*2−*x*2−*q*3*, x*−*y*2*, y*2*x, y*2−*q*4*,* −*y*2−*q*5*,*
where *q*1*, . . . , q*5∈R*.*

*Definition 3.3. We say that a complex polynomial is totally imaginary if the real part of all its*
coeﬃcients is zero.

* Lemma 3.4. Letg*∈AR

_{2}

_{n}_{−}

_{4}

*be a polynomial with real coeﬃcients. Thengis a real Hessian polynomial*

*if and only if there exists a polynomialftotally imaginary on the set*

*H*C−1−

_{n}*g.*

1By Lemma3.4and Proposition2.9we have that the polynomials *y*2_{−}* _{x}*2

_{−}

*1*

_{r}*,*−

*y*2−

*x*2_{−}_{r}

2*,*with*r*1*, r*2*>*0 are real Hessian polynomials. To finish this part we note that
the polynomial*fx, y* *xy*2_{/}_{2}_{}* _{x}*2

_{/}_{2}

_{∈}

_{A}R

3 verifies *H*3R*f* *x*−*y*2 and that

2By Lemma3.4and Proposition2.9we have that the polynomials −*x*2_{}* _{y}*2

_{,}_{−}

*2*

_{x}_{−}

*y*2* _{, y}*2

_{−}

*2*

_{x}_{−}

_{r}3*, y*2*x*2−*r*4*,* −*y*2−*x*2*r*5*,*with*r*3*, r*5*<*0*, r*4∈R*,*are not real Hessian
polynomials. On the other hand, the polynomials *y*2_{−}_{r}

6*,* −*y*2−*r*7*,*where *r*6*, r*7∈R∗*,*
are not real Hessian polynomials because the complex polynomial *y*2_{}_{r, r}_{∈}_{C}∗_{is}

not complex Hessian polynomial. To show that the polynomials *y*2and *y*2*x*are
not real Hessian polynomials, we use the same method of Groebner basis realized
in the complex case.

*Proof of Lemma3.4.*⇒By hypothesis there exists*fx, y* *n _{r}*

_{}

_{s}_{}

_{0}

*arsxrys*∈ AR

*n*such that

*H _{n}*R

*f*

*g*. Therefore, consider the totally imaginary polynomial

*ifx, y*

_{r}n_{}

_{s}_{}

_{0}

*iars*

*xrys*,

which satisfies*H _{n}*C

*if*

*i*2

*C*

_{H}*nf* −*g*.

⇐By hypothesis there exists*f*totally imaginary such that*H _{n}*C

*f*−

*g*. Therefore,

*if*

is a real polynomial such that*H _{n}*R

*if*

*i*2

*C*

_{H}*nf* *g.*Hence,*g*is a real Hessian polynomial.

*Proof of Corollary2.8. Let us consider the map* *ψ*:AC_{3} ×C2 _{→} _{C}3_{}_{{}_{}_{w}

1*, w*2*, w*3} given by

*f, p*−→*fxx*

*p, fxy*

*p, fyy*

*p.* 3.35

If*fx, y* 3_{i}_{}_{j}_{}_{0}*aijxiyj*∈AC3 and*p* *x, y*, then

*ψf, p*6*a*30*x*2*a*21*y*2*a*20*,*2*a*21*x*2*a*12*y*2*a*11*,*2*a*12*x*6*a*03*y*2*a*02

*.* 3.36

For each fixed *f*∈AC_{3} let us consider *ψf* :C2 → C3 given by *ψfp*:*ψf, p*.

Now, we are interested to describe the conditions in*f* under which the image under

*ψf* of C2*, ψf*C2, is not a plane.

**Lemma 3.5. The set***Sψ*:{*f*∈AC_{3} |*ψf*C2 *is not a plane*} *is given by the union of the following*

*sets:*

*CP*1:

*f*∈AC_{3} |*a*30 *a*21*a*120

*,*

*CP*2:

*f*∈AC_{3} |*a*30∈C∗*, a*12

*a*2
21
3*a*30*, a*03

*a*3_{21}

27*a*2_{30}

*.*

3.37

*Proof. The Jacobian matrix ofψf* is

⎛

⎝3_{a}a_{21}30 *a _{a}*21

_{12}

*a*12 3*a*03
⎞

⎠_{.}_{}_{3.38}_{}

*ψf*C2is not a plane if and only if *Jψf* has not maximal rank, which is equivalent to solve

the system of equations given by the three minor equal to zero. That is,

3*a*12*a*30−*a*2210*,* 3*a*21*a*03−*a*122 0*,* 9*a*03*a*30−*a*21*a*12 0*.* 3.39

* Lemma 3.6. Letfx, y* 3

_{i}_{}

_{j}_{}

_{0}

*aijxiyj*

*be a complex polynomial such that*

*f*∈

*Sψ.*

*Case 1. Suppose* *f*∈*CP*1.

1If*a*03*/*0*,*then *ψf*C2is a parallel line to the*w*3-axis inC3.

2If*a*030*,*then *ψf*C2is the point2*a*20*, a*11*,*2*a*02.

*Case 2. If* *f*∈*CP*1*,*then *ψf*C2is the line

*lf*

6*a*30*x*2*a*20*,*2*a*21*xa*11*,*
2*a*2

21

3*a*30*x*2*a*02

|*x*∈C

*.* 3.40

*Proof. Letfx, y* 3_{i}_{}_{j}_{}_{0}*aijxiyj*∈AC_{3}*.*In virtue of Lemma3.5we have the following cases

*Case 1. If* *f*∈*CP*1*,*then, by3.36, *ψf*C2 {2*a*20*, a*11*,*6*a*30*y*2*a*02∈C3|*y*∈C}*.*

1If*a*03*/*0*,*then *ψf*C2 {2*a*20*, a*11*,*6*a*30*y*2*a*02∈C3|*y*∈C}*.*

2If*a*030*,*then *ψf*C2is the point2*a*20*, a*11*,*2*a*02.

*Case 2. If* *f*∈*CP*2*,*then, by3.36,

*ψf*

C2_{}

6*a*30*x*2*a*21*y*2*a*20*,*2*a*21*x*
2*a*2

21
3*a*30

*ya*11*,*
2*a*3

21
9*a*2

30

*y*2*a*

2
21
3*a*30

*x*2*a*02

*.* 3.41

Let *v*1 6*a*30*,*2*a*21*,*2*a*212 */*3*a*30 be a nonzero vector by hypothesis and *v*2

2*a*21*,*2*a*21*/*3*a*30*,*2*a*3_{21}*/*9*a*2_{30}. Note that*v*2 *a*21*/*3*a*30*v*1. Therefore, the set*lf* is the same set

of *ψf*C2.

Let us consider the cone *C*{*w*1*, w*2*, w*3∈C3 |*w*1*w*3−*w*_{2}2 0}*.*We shall describe
the set *ψf*C2∩*C* when *f*∈*Sψ*.

**Lemma 3.7. Let***f* ∈*Sψ.*

1*If* *ψf*C2*is a parallel line to thew*3*-axis in*C3*, then*

a*ψf*C2∩*C* *is a parallel line to thew*3*-axis in*C3*whenevera*20 *0;*

b*ψf*C2∩*C* *is the point*2*a*20*, a*11*,a*112 −4*a*20*a*02*/*2*a*202*a*20 *whenevera*20*/0.*

2*If* *ψf*C2*is the point*2*a*20*, a*11*,*2*a*02 *and 4a*20*a*02−*a*211 *0, then* *ψf*C2 ∩ *Cis the*

*point*2*a*20*, a*11*,*2*a*02*.*

3*If* *ψf*C2*is the linelf, then*

a*ψf*C2∩*C* *is the point*6*a*30*x*02*a*20*,*2*a*21*x*0*a*11*,*2*a*2_{21}*/*3*a*30*x*02*a*02*ifα*

1*/*3*a*3036*a*2_{30}*a*024*a*20*a*2_{21}−12*a*21*a*11*a*30*/*0*,where* *x*0 1*/αa*2_{11}−4*a*20*a*02*;*

*Proof. In virtue of Lemma*3.6we have the following cases.

1If *ψf*C2∩*C* is a parallel line to the*w*3-axis, then *ψf*C2∩*C* {2*a*20*, a*11*,*6*a*03*y*
2*a*02∈C3|*y*∈C*,*2*a*206*a*03*y*2*a*02−*a*2_{11} 0}*.*

aIf*a*20 0, then *ψf*C2∩*C* is a parallel line to the*w*3-axis.

bIf*a*20*/*0, then*y* *a*2_{11}−4*a*20*a*02*/*12*a*206*a*03and *ψf*C2∩*C* is a point.

2If *ψf*C2∩*C* is the point2*a*20*, a*11*,*2*a*02, then *ψf*C2∩*C* 2*a*20*, a*11*,*2*a*02if
4*a*20*a*02−*a*211 0.

3If *ψf*C2∩*C* is the line*lf*, then *ψf*C2∩*C* *lf*∩ {6*a*30*x*2*a*202*a*2_{21}*/*3*a*30*x*
2*a*20−2*a*21*xa*1120}. It means that

*αx*4*a*20*a*02−*a*211 0*,* where*α*

36*a*2_{30}*a*024*a*20*a*2_{21}−12*a*21*a*11*a*30

3*a*30 *.*

3.42

Therefore, if*α*0, then 4*a*20*a*02−*a*2110 and we obtain a line on*C*. If*α /*0, then *ψf*C2∩*C*

is a point.

With this lemma we finish the proof of corollary.

*Proof of Proposition2.9. Letgx, y* *b*20*x*2 *b*11*xyb*02*y*2 *b*00 ∈ AC_{2}. Then the expression

*H*_{3}C*f* *g*is equivalent, by Lemma3.1, to the system of equations:

*b*02−4*a*2_{21}12*a*30*a*12*,*

*b*11−4*a*21*a*1236*a*30*a*03*,*

*b*02−4*a*2_{12}12*a*21*a*03*,*

012*a*30*a*024*a*20*a*12−4*a*21*a*11*,*

012*a*20*a*034*a*21*a*02−4*a*12*a*11*,*

*b*00−*a*2114*a*20*a*02*.*

3.43

Let*S*be the union of the images of*F*1and*F*2. A direct substitution shows that*S*⊂*H*3C−
1_{}_{g}_{}_{.}

Therefore, to finish the proof it is enough to show that *H*_{3}C−1*g* ⊂ *S*. To check this last
sentence we have used the computer algebra system Maple 9.5.

*Proof of the Theorem2.12.*

*Real Case*

real cuartic polynomial*f* in *H*_{4}C−1*g*; that is, there are no real polynomial satisfying the
system of equations:

24*a*40*a*22−9*a*2_{31} 0*,* 3.44

72*a*40*a*13−12*a*31*a*220*,* 3.45

144*a*40*a*0418*a*31*a*13−12*a*2220*,* 3.46

72*a*31*a*04−12*a*22*a*130*,* 3.47

24*a*04*a*22−9*a*2130*,* 3.48

12*a*30*a*2224*a*40*a*12−12*a*31*a*21 −1*,* 3.49

36*a*30*a*1372*a*40*a*03−12*a*21*a*220*,* 3.50

36*a*03*a*3172*a*04*a*30−12*a*12*a*221*,* 3.51

12*a*03*a*2224*a*04*a*21−12*a*13*a*120*,* 3.52

24*a*40*a*024*a*20*a*2212*a*30*a*12−6*a*31*a*11−4*a*2210*,* 3.53

12*a*31*a*0212*a*20*a*1336*a*30*a*03−4*a*21*a*12−8*a*11*a*22 0*,* 3.54

24*a*04*a*204*a*02*a*2212*a*03*a*21−6*a*13*a*11−4*a*2120*,* 3.55

12*a*30*a*024*a*20*a*12−4*a*21*a*11 1*,* 3.56

12*a*03*a*204*a*02*a*21−4*a*12*a*11 0*,* 3.57

4*a*20*a*02−*a*2110*.* 3.58

A Groebner bases for3.44–3.48with the graded reverse lexicographic monomial order is

3*a*2_{13}−8*a*22*a*04*, a*22*a*13−6*a*31*a*04*,* 2*a*222 −3*a*31*a*13−24*a*40*a*04*, a*31*a*22−6*a*40*a*13*,*

3*a*2

31−8*a*40*a*22*, a*13*a*31*a*04−16*a*40*a*204*, a*31*a*40*a*13−16*a*240*a*04*.*

3.59

The set of common zeroes of these last 7 polynomials is the union of the sets

{*a*40*a*40*, a*310*, a*220*, a*13 0*, a*04 0}*,* 3.60

*a*40

*a*4
13
256 *a*3

04

*, a*31

*a*3
13
16*a*2

04

*, a*22
3*a*2

13
8*a*04

*, a*13 *a*13*, a*04*/*0

On the one hand, replacing the solution3.60in3.49–3.58, particularly in3.56we obtain 01. On the other hand, replacing the solution3.61in3.49–3.58, we obtain the system

4*a*20*a*02−*a*2110*,*

12*a*30*a*024*a*20*a*12−4*a*21*a*11−10*,*

−4*a*12*a*114*a*21*a*0212*a*20*a*030*,*

12*a*30*a*12
3*a*4_{13}*a*02

32*a*3_{04} −4*a*

2 21−

3*a*3_{13}*a*11
8*a*2

04

3*a*20*a*213
2*a*04

0*,*

36*a*30*a*0312*a*20*a*13−
3*a*2

13*a*11

*a*04 −

4*a*21*a*12
3*a*3

13*a*02
4*a*2

04

0*,*

−6*a*13*a*1124*a*20*a*04
3*a*2_{13}*a*02

2*a*04 12*a*21*a*03−4*a*
2
12 0*,*

−3*a*3_{13}*a*21
4*a*2_{04}

9*a*30*a*2_{13}
2*a*04

3*a*4_{13}*a*12

32*a*3_{04} 10*,*

9*a*4
13*a*03
32*a*3

04

36*a*30*a*13−

9*a*21*a*2_{13}
2*a*04

0*,*

9*a*3
13*a*03
4*a*2

04

−9*a*213*a*12
2*a*04

72*a*30*a*04−10*,*

−12*a*13*a*12

9*a*2_{13}*a*03
2*a*04

24*a*21*a*040*.*

3.62

From the last equation we obtain that *a*21 −*a*13−8*a*12*a*04 3*a*13*a*03*/*16*a*2_{04} and from the
sixth equation,

*a*20 −
6*a*2

13*a*02*a*04−24*a*13*a*11*a*20424*a*13*a*03*a*12*a*04−9*a*213*a*203−16*a*212*a*204

96*a*3_{04} *.* 3.63

When we replace these expressions in the last four equations of the last systemwe do not write the other equations because we do not need it:

−18*a*4

13*a*12*a*049*a*513*a*03288*a*30*a*213*a*30464*a*404
64*a*4

04

0*,* 3.64

9*a*13

32*a*30*a*3_{04}−2*a*12*a*2_{13}*a*04*a*3_{13}*a*03

8*a*3
04

9*a*3_{13}*a*03−18*a*12*a*2_{13}*a*04288*a*30*a*3_{04}−4*a*2_{04}

4*a*2_{04} 0*,* 3.66

00*.* 3.67

From3.65we have that *a*130 or 32*a*30*a*3_{04}−2*a*12*a*_{13}2 *a*04*a*3_{13}*a*03 0. If *a*13 0*,*then3.64
becomes −10*.*

If 32*a*30*a*3_{04}−2*a*12*a*_{13}2 *a*04*a*3_{13}*a*030*,*then *a*30 2*a*12*a*2_{13}*a*04−*a*3_{13}*a*03*/*32*a*3_{04}. Substituing

*a*30in3.66, we obtain−10.

* Lemma 3.8. The set,Cv*3

*, of critical points of the map*

*H*

_{3}C

*:*

_{R}*A*C

_{3}

*→*

_{R}*A*C

_{2}

*is the union of the*

*following six sets:*

*S*1

*a*02

9*a*03*a*11*a*30−6*a*03*a*21*a*20−*a*12*a*11*a*212*a*2_{12}*a*20
23*a*12*a*30−*a*2_{21}

*,*3*a*12*a*30−*a*221*/*0

*,*

*S*2{*a*110*, a*210*, a*120}*,*

*S*3

*a*11

2*a*12*a*20

*a*21

*, a*30

*a*2_{21}

3*a*12

*, a*21*a*12*/*0

*,*

*S*4{*a*300*, a*210*, a*120}*,*

*S*5{*a*200*, a*300*, a*210}*,*

*S*6

*a*12

*a*2
21
3*a*30

*, a*03

*a*3
21
27*a*2
30

*, a*30*/*0

*.*

3.68

*Proof. Letfx, y* 3_{i}_{}_{j}_{}_{2}*aijxiyj*∈AC_{3}* _{R}*. The Jacobian matrix of

*H*

_{3}C

*is,*

_{R}*JH*_{3}C_{R}

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

2*a*02 −*a*11 2*a*20 0 0 0 0

6*a*03 −2*a*12 2*a*21 0 2*a*02 −2*a*11 6*a*20

2*a*12 −2*a*21 6*a*30 6*a*02 −2*a*11 2*a*20 0

0 0 0 0 6*a*03 −4*a*12 6*a*21

0 0 0 18*a*03 −2*a*12 −2*a*21 18*a*30

0 0 0 6*a*12 −4*a*21 6*a*30 0
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠

Let us denote by *Mk, k* 1*, . . . ,*7*,*the 6×6 matrix obtained from *JH*_{3}C* _{R}*by deleting the

column 7−*k*1. On the other hand,*f*is a critical point of*H*_{3}C* _{R}*if and only if det

*Mk*0 for

all*k*1*, . . . ,*7, that is, if and only if the following system of seven equations is satisfied:

−9*a*21*a*12*a*0327*a*30*a*2032*a*312

*F* 0*,*

*a*2_{12}*a*21−6*a*221*a*039*a*12*a*30*a*03

*F* 0*,*

*a*2_{21}*a*12−6*a*212*a*309*a*21*a*30*a*03

*F* 0*,*

2*a*3_{21}−9*a*12*a*21*a*3027*a*230*a*03

*F* 0*,*

−3*a*03*a*21*a*11*a*212*a*11−*a*21*a*12*a*029*a*03*a*30*a*02

*F* 0*,*

3*a*02*a*12*a*30*a*212*a*20−3*a*03*a*21*a*20−*a*221*a*02

*F* 0*,*

*a*11*a*2_{21}−*a*12*a*21*a*209*a*20*a*30*a*03−3*a*12*a*11*a*30

*F* 0*,*

3.70

where

*F*−9*a*03*a*11*a*306*a*03*a*21*a*206*a*02*a*12*a*30−2*a*221*a*02*a*12*a*11*a*21−2*a*212*a*20*.* 3.71

Therefore*fx, y* 3_{i}_{}_{j}_{}_{2}*aijxiyj*satisfies the previous system of seven equations if and only

if*f* ∈*S*6_{k}_{}_{1}*Sk*, where*S*is the union of the six solutions*Sj*. Therefore, the proof is done.

* Lemma 3.9. Considerf* ∈AC

_{3}

*and*

*g*

*H*3

*f. The curve*

*g*

*0 is singular if and only if*

−2*a*2_{12}*a*206*a*12*a*02*a*306*a*21*a*03*a*20*a*21*a*12*a*11−2*a*2_{21}*a*02−9*a*03*a*11*a*300*,*

−*a*2_{21}*a*2_{12}−18*a*21*a*12*a*30*a*0327*a*302 *a*2034*a*312*a*304*a*321*a*03*/*0*,*

3.72

*or,*

−9*a*12*a*21*a*03*a*20−*a*21*a*11*a*212−*a*12*a*212 *a*0227*a*30*a*203*a*202*a*20*a*3121*.*8

6*a*2_{21}*a*03*a*11−9*a*30*a*21*a*03*a*026*a*30*a*02*a*212−9*a*30*a*03*a*11*a*12 01*,*

−*a*2_{21}*a*2_{12}−18*a*21*a*12*a*30*a*0327*a*2_{30}*a*2_{03}4*a*3_{12}*a*304*a*3_{21}*a*03 0*.*2*.*1*,*

*Proof. Supposefx, y* 3_{r}_{}_{s}_{}_{0} *arsxrys*; then

*H*3

*f*−*a*2

114*a*20*a*02 12*a*30*a*024*a*20*a*12−4*a*21*a*11*x*

12*a*20*a*03−4*a*12*a*114*a*21*a*02*y*

−4*a*2_{21}12*a*30*a*12

*x*2

−4*a*21*a*1236*a*30*a*03*yx*

−4*a*2_{12}12*a*21*a*03

*y*2*.*

3.74

The curve *g* *H*_{3}C*f*is singular if and only if the system *gxp* *gyp* *gp* 0 has a

solution for some*p*inC2_{. The system formed by}_{g}

*x*0*, gy*0 is

6*a*30*a*12−2*a*221

*x* 9*a*30*a*03−*a*21*a*12*y*3*a*30*a*02*a*20*a*12−*a*21*a*110*,*

9*a*30*a*03−*a*21*a*12*x*

6*a*21*a*03−2*a*212

*y*3*a*20*a*03−*a*12*a*11*a*21*a*020*.*

3.75

The proof concludes by analyzing the system when its determinant is distinct from zero and when it is zero.

*Proof of Proposition2.10. Let us consider the setsSk, k*1*, . . . ,*6*,*of Lemma3.8. If*f*∈*S*1, then

*H*_{3}C*f* 1

3*a*12*a*30−*a*2_{21}

×*x*236*a*2_{30}*a*2_{12}4*a*4_{21}−24*a*30*a*12*a*221

*xy*108*a*2

30*a*03*a*1236*a*30*a*03*a*2214*a*321*a*12−12*a*21*a*212*a*30

*y*2_{4}* _{a}*2

12*a*221−12*a*312*a*3036*a*21*a*03*a*12*a*30−12*a*321*a*03

*x*24*a*30*a*212*a*2054*a*230*a*03*a*114*a*3_{21}*a*11−4*a*20*a*12*a*221

−36*a*30*a*03*a*21*a*20−18*a*30*a*12*a*11*a*21

*y*−12*a*2_{12}*a*11*a*3018*a*21*a*03*a*11*a*3036*a*20*a*03*a*12*a*30

−24*a*2_{21}*a*03*a*202*a*_{21}2 *a*12*a*114*a*21*a*2_{12}*a*20

−3*a*2_{11}*a*12*a*30−12*a*03*a*21*a*22018*a*20*a*03*a*11*a*30

−2*a*20*a*12*a*11*a*214*a*212*a*220*a*211*a*221

*.*

Denote by*h*the polynomial*H*_{3}C*f*. Then,

*hx* 1

3*a*12*a*30−*a*2_{21}

×*x*72*a*2_{30}*a*2_{12}−48*a*30*a*12*a*2_{21}8*a*4_{21}

*y*108*a*2_{30}*a*03*a*124*a*3_{21}*a*12−36*a*30*a*03*a*2_{21}−12*a*21*a*2_{12}*a*30

4*a*3_{21}*a*1154*a*230*a*03*a*11−4*a*20*a*12*a*2_{21}24*a*30*a*2_{12}*a*20

−36*a*30*a*03*a*21*a*20−18*a*30*a*12*a*11*a*21

*,*

*hy* 1

3*a*12*a*30−*a*2_{21}

×*x*108*a*2_{30}*a*03*a*12−36*a*30*a*03*a*221−12*a*21*a*212*a*304*a*3_{21}*a*12

*y*8*a*2_{12}*a*2_{21}−24*a*3_{12}*a*30−24*a*3_{21}*a*0372*a*21*a*03*a*12*a*30

4*a*21*a*212*a*20−24*a*221*a*03*a*202*a*221*a*12*a*11−12*a*212*a*11*a*30

18*a*21*a*03*a*11*a*3036*a*20*a*03*a*12*a*30

*.*

3.77

The point*x, y*where the Hessian curve is singular is

*x* −2*a*20*a*12*a*21*a*11

23*a*12*a*30−*a*2_{21}

*,* *y*−3*a*11*a*30−2*a*21*a*20

23*a*12*a*30−*a*2_{21}

*.* _{}3.78

If*f* ∈ *S*2, then *H*3C*f* 6*a*30*x*2*a*206*a*03*y*2*a*02*, hx* 6*a*306*a*03*y*2*a*02*,* and*hy*
66*a*30*x*2*a*20*a*03*.*

The point*x, y*where the Hessian curve is singular is *x*−*a*20*/*3*a*30*, y*−*a*02*/*3*a*03*.*
If*f*∈*S*3, then

*H*_{3}C*f* −4

*a*2_{21}*xa*21*ya*12*a*20*a*12

*y*−3*a*2_{21}*a*03*a*21*a*2_{12}

−*a*2_{21}*a*02*a*20*a*2_{12}

*a*12*a*2_{21}

*,*

*hx* −4

*y*−3*a*2_{21}*a*03*a*21*a*2_{12}

−*a*2_{21}*a*02*a*20*a*2_{12}

*a*12

*,*

*hy*−4

*y*−3*a*2

21*a*03*a*21*ya*212

−*a*2

21*a*02*a*20*a*212

*a*21

− 4

*a*2_{21}*xa*21*ya*12*a*20*a*12

−3*a*2_{21}*a*03*a*21*a*2_{12}

*a*12*a*2_{21}

*.*

The point*x, y*where the Hessian curve is singular is

*x* *a*12−*a*21*a*023*a*20*a*03
*a*21

−3*a*21*a*03*a*2_{12}

*,* *y*−−

−*a*2_{21}*a*02*a*20*a*2_{12}

*a*21

−3*a*21*a*03*a*2_{12}

*.* 3.80

If−3*a*21*a*03*a*_{12}2 0, then*H*_{3}C*f*is of degree one.

If*f*∈*S*4, then *H*_{3}C*f* 12*a*20*a*03*y*4*a*20*a*02−*a*2_{11}*, hx*0*,* and*hy*12*a*20*a*03*.*
The Hessian curve in this case has degree one.

If*f*∈*S*5, then *H*_{3}C*f* −2*a*12*ya*112*, hx*0*, hy*−42*a*12*ya*11*a*12*.*
The point*x, y*where the Hessian curve is singular is *y*−*a*11*/*2*a*12*.*
If*f*∈*S*6, then

*H*_{3}C*f*

108*a*3_{30}*a*0212*a*20*a*_{21}2 *a*30−36*a*21*a*2_{30}*a*11
9*a*2

30

*x*

36*a*21*a*02*a*2_{30}4*a*20*a*3_{21}−12*a*2_{21}*a*11*a*30
9*a*2

30

*y*36*a*20*a*02*a*

2

30−9*a*211*a*230
9*a*2

30

*.*

3.81

The Hessian curve in this case has degree one.

**Acknowledgments**

The authors would like to thank L. Ortiz-Bobadilla, E. Rosales-Gonz´alez, and R. Uribe-Vargas for their interesting comments.

**References**

1 *G. Salmon, A Treatise on the Analytic Geometry of Three Dimensions, vol. 2, Chelsea, New York, NY, USA,*
5th edition, 1965.

2 *Y. L. Kergosien and R. Thom, “Sur les points paraboliques des surfaces,” Comptes Rendus Hebdomadaires*
*des S´eances de l’Acad´emie des Sciences. S´eries A, vol. 290, no. 15, pp. 705–710, 1980.*

3 V. I. Arnold, “Remarks on parabolic curves on surfaces and on higher-dimensional M ¨obius-Sturm
*theory,” Funktsional’nyi Analiz i ego Prilozheniya, vol. 31, no. 4, pp. 3–18, 1997* Russian, english
*translation in Functional Analysis and Its Applications, vol. 31, no. 4, pp. 227–239, 1997.*

4 *V. I. Arnold, Arnold’s Problems, Springer, Berlin, Germany, 2004.*

5 *V. I. Arnold, “On the problem of realization of a given Gaussian curvature function,” Topological*
*Methods in Nonlinear Analysis, vol. 11, no. 2, pp. 199–206, 1998.*

6 *E. Brieskorn and H. Kn ¨orrer, Plane Algebraic Curves, Birkh¨auser, Basel, Switzerland, 1986.*