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Lomas de Zaragoza C. Circuito Exterior, C. 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. 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, Salmon [ 1 ], Kergosien and Thom [ 2 ], Arnold [ 3 ], among others.

The parabolic curve of the graph of a smooth function, , is the set : , where Hess. In this case, the Hessian curve of , Hess , is a plane curve which is the projection of the parabolic curve into the -plane along the -axis.

When is a polynomial of degree in two variables, Hess is a polynomial of degree at most. 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.

Given an algebraic plane curve in , where or , we ask: When is the Hessian curve of a polynomial? If such exists and , then is called a complex Hessian polynomial real Hessian polynomial. We remark that Arnold see [ 4 ] calls Hessian topology problem to the study of the problem 1 in the real case. 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. In the second part, we give the proofs and some other results such as a geometric interpretation of Corollary 2. For each nonnegative integer number we define. Let us consider the Hessian map. We will say that the Hessian map is complex or real if or , respectively.

We remark that dim dim if and dim dim if. In particular, for , the image, is a connected subset of codimension at least three in. In diagram form we have. In virtue of the previous remarks, we introduce the fiber of under as the set Even when we consider the fibers, , for different values of , we are interested in knowing if the fiber is not empty when satisfies If the fiber is empty, the next problem is to see if the fibers are not empty for 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 , with such that whenever. Remark 2. Let be a polynomial in. There exists a polynomial in if and only if satisfies the following system of equations: where are the coefficients of the Hessian polynomial. They are also quadratic polynomials in the variables. It is important to note that the computations for solving the system 2. 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.

Proposition 2. For each , the set is a quadric in given by Moreover, this quadric is singular if and only if. Corollary 2. In the complex case every element in is a complex Hessian polynomial under. And, in the real case every element in is a real Hessian polynomial under.

For each of degree one, the set is an analytic subvariety in which is given by the union of analytic subvarieties parametrized by the following. Every complex polynomial of degree one is a complex Hessian polynomial under. And every real polynomial of degree one is a real Hessian polynomial under. We say that is in the orbit of if they are equivalent by an affine transformation of the plane. Theorem 2. Complex Case The complex polynomials of degree two are complex Hessian polynomials under.

Moreover, a polynomial of degree two is a complex Hessian polynomial under if and only if it belongs to the orbit of one of those polynomials. The complex polynomials, where are not complex Hessian polynomials under. Moreover, a polynomial of degree two is not a complex Hessian polynomial under if and only if it belongs to the orbit of one of those polynomials.

Real case 1 The real polynomials of degree two are real Hessian polynomials under. Moreover, a polynomial of degree two is not a real Hessian polynomial under 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: parabola , general conic , line pair , parallel lines , and finally double line.

Therefore, we have the following corollary. All the complex affine conics, except the parallel lines, are complex Hessian curves of polynomials in. Let be a polynomial in with. Then the set is an analytic subvariety on which is the union of analytic subvarieties parametrized by the following. From the study of the previous fibers, a natural question arises. We define. Let us describe the relation between the set of critical points of and the set of polynomials in such that they define singular Hessian curves.

If is a critical point of the map , then the Hessian curve is singular or it has degree one. Conjecture 2. If is a critical point of the map , then its Hessian curve, Hess is singular or has degree less than. Complex Case 1 The curves defined by Table 1 of complex cubic curves see [ 6 ] are complex Hessian curves under. Real Case 1 The normal form of curves see [ 7 ] defined in Table 3 is a real Hessian curve under. Moreover, a curve of degree three is a real Hessian curve under if and only if its polynomial belongs to the orbit of one of those polynomials.

Let us consider the complex Hessian map and recall that the fiber of under is the set. A direct calculus shows that , and. Therefore, For each , consider. To show that it is enough to show because a direct substitution shows. Therefore, let us consider , that is,. Hence and the first part of the claim is done.

Finally, the derivative of is given by Therefore, we conclude the proof of the result. The polynomial satisfies that in complex or real case. Lemma 3. If , then the map is where the coefficients satisfy the following system of quadratic equations:. For we have that, by Lemma 3. To prove that , it is enough to show that because a direct substitution shows that.

Now, to prove we will consider two cases: Case 1 is when and Case 2 is when. Case 1. In this case, from a direct substitution we obtain , , , and 3. To solve these equations we will assume the following. Subcase 1. In this case we obtain ;. All this values together are contained in the set whose parametrization is. This case will be subdivided in two subcases. First, we obtain from 3. Later, from a substitution of together with the value of in 3.

This equation implies. All these values together are contained in the set whose parametrization is. In this case, from 3.

Finally, from a substitution of in 3. From these values we obtain the parametrization. Case 2. From a direct calculus we obtain and the equations: To solve these four equations we will consider two cases. Subcase 2. From a direct substitution we get and the two equations: To solve these two equations we will consider two subcases.

We obtain and. From all of these values we get the parametrization. We obtain. From all these values we get the parametrization when.

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## On the Complex and Real Hessian Polynomials

Lomas de Zaragoza C. Circuito Exterior, C. 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. 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, Salmon [ 1 ], Kergosien and Thom [ 2 ], Arnold [ 3 ], among others. The parabolic curve of the graph of a smooth function, , is the set : , where Hess. In this case, the Hessian curve of , Hess , is a plane curve which is the projection of the parabolic curve into the -plane along the -axis.

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