Plane polynomials and Hamiltonian vector fields determined by their singular points

Let $\Sigma(f)$ be critical points of a polynomial $f \in \mathbb{K}[x,y]$ in the plane $\mathbb{K}^2$, where $\mathbb{K}$ is $\mathbb{R}$ or $\mathbb{C}$. Our goal is to study the critical point map $\mathfrak{S}_d$, by sending polynomials $f$ of degree $d$ to their critical points $\Sigma(f)$ . Very roughly speaking, a polynomial $f$ is essentially determined when any other $g$ sharing the critical points of $f$ satisfies that $f= \lambda g$; here both are polynomials of at most degree $d$, $\lambda \in \mathbb{K}^*$. In order to describe the degree $d$ essentially determined polynomials, a computation of the required number of isolated critical points $\delta (d)$ is provided. A dichotomy appears for the values of $\delta (d)$; depending on a certain parity the space of essentially determined polynomials is an open or closed Zariski set. We compute the map $\mathfrak{S}_3$, describing under what conditions a configuration of four points leads to a degree three essentially determined polynomial. Furthermore, we describe explicitly configurations supporting degree three non essential determined polynomials. The quotient space of essentially determined polynomials of degree three up to the action of the affine group $\hbox{Aff}(\mathbb{K}^2)$ determines a singular surface over $\mathbb{K}$.


Introduction
Let K " R or C. We then ask under what conditions a polynomial f P Krx, ys is essentially determined by its critical points Σpf q Ă K 2 ?Thus, we want to study the critical point map sending polynomials of degree d to their critical points where Σpf q ." Ipf x , f y q is the affine algebraic variety (not necessarily reduced) generated by the ideal of partial derivatives of f , see Definition 7. Our approximation route uses a finite dimensional framework.Let Krx, ys 0 ďd be the K-vector space of polynomials having at most degree d (ě 3) and zero independent term, and let P " tpx ι , y ι qu be a configuration of n different points in the plane.The linear projective subspace of the polynomials with critical points at least in P, denoted as is well defined.We say that a polynomial f is essentially determined by P when L d pPq is a projective point tλf | λ P K ˚u, see Definition 4. All this leads us to the following.
Interpolation problem for critical points.Let P Ă K 2 be a configuration of n different points, we try to determine the projective subspace L d pPq of polynomials of at most degree d with critical points at least in P.
This problem has several novel features.The critical values tc ι u Ă K of f can appear in different level curves tf px, yq ´cι " 0u; it is natural in Hamiltonian vector field theory and moduli spaces of polynomials, see P. G. Wightwick [18] and J. Fernández de Bobadilla [11].This is a main difference with the widely considered problem of linear system of curves in CP 2 , e.g.R. Miranda, [15] and C. Ciliberto [8].
Very roughly speaking, for degree d ě 3 the relevant data are the cardinality and position of the configuration P, as candidate to be a critical point configuration Σpf q.For degree three, the prescription of four critical points is suitable.For degree d ě 4, however, the generic configuration P having pd ´1q 2 points is too restrictive.Thus the fiber S ´1 d pPq will be generically empty.It follows that, the position of the configurations P coming from polynomials is the hardest part to be characterized.At this first stage, we consider mainly P as isolated points of multiplicity one, Remark 1 provides an explanation.Our first result describes the role of cardinality δpdq of P in Eq. (2), see Proposition 1.
Dichotomy on the required number of critical points If the dimension of Krx, ys 0 ďd is odd (resp.even) then the configurations tPu with δpdq points and dim K pL d pPqq ě 0 determine an open (resp.closed) Zariski set in the space of configurations with δpdq points, denoted as Conf pK 2 , δpdqq.
We compute the critical point map S 3 .Thus, a description for the four critical point configurations tPu with essentially determined polynomials is provided.Recall that the affine group Aff pK 2 q acts on the space of polynomials, see Eq. ( 20).This action is rich enough and yet treatable for degree three.Let

A
.
Theorem 1.Let f be a degree three polynomial having at least four critical points Σpf q. 1) f is essentially determined if and only if up to affine transformation the four points are Σpf q " tp0, 0q, p1, 0q, p0, 1q, px 4 , y 4 qu and px 4 , y 4 q R A .
2) f is not essentially determined if and only if up to affine transformation the four points are tp0, 0q, p1, 0q, p0, 1q, px 4 , y 4 qu and px 4 , y 4 q P A .
Moreover, in this case Σpf q can be four isolated points or two parallel lines.
In simple words, the 4-th point px 4 , y 4 q generically determines the polynomial f .We compute the fundamental domain for this Aff pK 2 q-action, obtaining a tessellation of K 2 " tpx 4 , y 4 qu with 24 tiles, a seen in Fig. 3.
As is expected, some interesting phenomena occur for configurations with non trivial isotropy groups.For degree d ě 3, a particular family of configurations is the grids of pd ´1q 2 points from the intersection of two families of d parallel lines in K 2 , see Definition 8.They provide examples of non essential determined polynomials with pd ´1q 2 Morse critical points.A remaining open question, are these grids of pd ´1q 2 points the unique mechanism in order to produce non essential determined Morse polynomials?
From the point of view of vector fields, we are studying under what conditions the zeros a Hamiltonian vector field determine it in a unique way?This is a very general and interesting issue in real and complex foliation theory, studied by; X. Gómez-Mont, G. Kempf [13], J. Artes, J. Llibre, N. Vulpe [4], A. Campillo, J. Olivares [6] and V. Ramírez [17] see Corollary 5. Related works are accurately described in Section §7.
The content of this work is as follows.In §2-3, we study the problem of the dimension of linear systems for polynomials with critical points, using the degree as parameter.In section §4, we characterize polynomials essentially determined by their configurations of critical points; this proves Theorem 1.In section §5, we focus in the degree four case.For each configuration of six points, we obtain a plane curve of degree six parametrizing the essentially determined polynomials, see Proposition 2. Section §6 explores the behavior of pencils of Hamiltonian vector fields with common simple zeros.

Linear systems L d pPq
Let Krx, ys 0 ďd (resp.Krx, ys 0 "d ) be the K-vector space of polynomials having at most degree d ě 3 (resp.the set for degree " d) and zero independent term.Consider from which the K-dimension of Krx, ys 0 ďd is 1 2 pd 2 `3dq, and its projectivization is where r s denotes a projective class.Recall that Conf pK 2 , nq " P " tpx 1 , y 1 q, . . ., px n , y n qu | px ι , y ι q ‰ px j , y j q for ι ‰ j ( {Sympnq is the space of unordered configurations of n points in K 2 , where the symmetric group in n elements, Sympnq, acts by exchanging the points.The configuration space Conf pK 2 , nq is a K-analytic manifold.
Definition 1.Given a configuration P P Conf pK 2 , nq, the linear system of polynomials of at most degree d with critical points at least in P is the projective subspace In algebraic geometry language, tf x px, yq " 0u, tf y px, yq " 0u belong to the linear system of algebraic curves L d´1 `´Σ n α"1 px ι , y ι q see [15] and [8].In several places however, we consider f x , f y as functions and not just as algebraic curves.The polynomials of at most degree d, the polynomial Hamiltonian vector fields and the polynomial vector fields, of at most degree d ´1, are related by linear maps In the space of Hamiltonian vector fields, L d pPq determines a linear subspace set theoretically, the zeros ZpλX f q of the vector field X f coincide with tf x px, yq " 0u X tf y px, yq " 0u.
Definition 2. Let f P Krx, ys be a non constant polynomial.Over K " C, the Milnor number of X f at a zero point px ι , y ι q P ZpXq is µ pxι,yιq pXq " dim C O C 2 ,pxι ,yι q ă´fy,fxą , where O C 2 ,pxι,yιq is the local ring of holomorphic functions at the point px ι , y ι q and ă ´fy , f x ą is the ring generated by the partial derivatives.
Remark 1. 1.Over K " C, if px ι , y ι q is an isolated singular point of f , then the notions of multiplicity for the intersection of the curves tf x px, yq " 0u Y tf y px, yq " 0u and the Milnor number for X f coincide; see [14] p. 174. 2. A priori, we consider each point px ι , y ι q P P in (6) with multiplicity of intersection one for the algebraic curves tf x px, yq " 0u and tf y px, yq " 0u. 3.By Bézout's theorem, the maximal number of isolated singularities of X f on C 2 is pd ´1q 2 .In this case all the affine singularities are of multiplicity one.4.Moreover, the maximal number of isolated singularities of X f extended to CP 2 is pd ´1q 2 `d.Here the upper bound d comes from the intersection of a generic projectivized level curve tf " cu with the line at infinity; see [13], [6] for the case of rational vector fields, which are not necessarily Hamiltonian.
Let A 2 K " Spec Krx, ys be the affine scheme of the affine plane K 2 , see [10] pp.48-49.Definition 3. The critical point map of degree d is the map sending a polynomial of degree d to its critical points Σpf q, as an affine algebraic variety (not necessarily reduced) generated by the ideal of partial derivatives of f .
In fact, Σpf q can be understood as a subscheme, with support at the points tf x px, yq " 0u X tf y px, yq " 0u, where the sheaf of ideals is defined by the germs of Ipf x , f y q; compare with [6], [10] p. 100.In a set theoretical language, Σpf q determines points and even algebraic curves.However in the study of rational vector fields on CP 2 , the case of foliations having singularities along curves is removed, see [13], [6].
Remark 2. The simplest case of the interpolation problem for singular points occurs when Σpf q is a finite set of points of multiplicity one, i.e. tf x px, yq " 0u and tf y px, yq " 0u have transversal intersections.The Σpf q is a configuration in Conf pK 2 , nq, for 0 ď n ď pd ´1q 2 .
Our former task is as follows: Given a configuration P, which is dim K pL d pPqq?To be clear, three relevant data must be considered the degree d of the polynomials tf u, the cardinality n and the position of the configuration P. The following diagram explains: The natural concepts are as follows.
Definition 4. Let f P Krx, ys 0 ďd be a polynomial and let P be a configuration of n points in K 2 .1) A polynomial f is essentially determined by P when rf s " L d pPq.
2) A polynomial f is non essential determined by P when rf s P L d pPq and dim K pL d pPqq ě 1.
3) P is a forbidden configuration (for polynomials of at most degree d) when L d pPq " H. 4) The set of degree d essentially determined polynomials is where the union is over all configurations tPu such that dim K pL d pPqq " 0.
Remark 3. 1) The strict set theoretical inclusion P Ł Σpf q can be satisfied for essentially determined polynomials f , for example as with the case of a product of three lines one with multiplicity two, say f " L 2 1 L 2 .
2) The set of degree three essentially determined polynomials E 3 is a union of projective spaces, however it is not a projective space, as Proposition 1 will show.3) As is expected, many of the projective classes in E d arise from Morse polynomials.The converse is not true, see Corollary 6.

On the number of required critical points
A novel aspect of the interpolation problem for critical points is its cardinality; the configurations having a certain number δpdq of points determine open or closed Zariski sets in Krx, ys 0 ďd .As a key point, the dimension 1 2 pd 3 `3dq of Krx, ys 0 ďd can be even or odd.Starting with degree d " 4, the pattern of these dimensions is 4-periodic; even, even, odd odd, . . . .See the third column in Table 1.
Proposition 1. (A dichotomy on the number δpdq of required critical points) Let Krx, ys 0 ďd be the set of polynomials having at most degree d ě 3 and let

δpdq
. " 1.If the dimension of Krx, ys 0 ďd is odd, then the configurations tPu with δpdq points and dim K pL d pPqq ě 0 determine an open Zariski set in Conf pK 2 , δpdqq.

If the dimension of Krx, ys 0
ďd is even, then the configurations tPu with δpdq points and dim K pL d pPqq ě 0 determine a closed Zariski set in Conf pK 2 , δpdqq.
Proof.Let f px, yq P Krx, ys 0 ďd be a polynomial as in (3).Assume that P " tpx ι , y ι q | ι " 1, . . ., nu is set theoretically contained in Σpf q.A priori, each point px ι , y ι q P P will drop the dimension of the vector space Krx, ys 0 ďd by two.In the linear framework, this leads to a linear system of 2n equations: with ta ιj u as variables.Following Bézout's theorem for a moment, let us consider a configuration with n " pd ´1q 2 points.We have a linear map The interpolation matrix φ depends on P, and by notational simplicity we omit this dependence.The matrix φ has 1 2 pd 2 `3dq columns, 2pd ´1q 2 rows and a very particular shape because of the partial derivatives involved in it, see Eqs. ( 17), (33) for explicit examples with d " 3, 4.
For degree d " 3 and a configuration P of 4 points; however then the rank of the matrix φ associated to P is 8 if and only if dim K pL 3 pPqq " 0. If we consider degree d ě 4, then the number of rows of φ is bigger than the number of columns.We must reduce the number n of required points in the configurations P, this n ă pd ´1q 2 .The number δpdq in (10) determines two possibilities.
Case 2 in (10).The dimension of Krx, ys 0 ďn is even and we assume 1 4 pd 2 `3dq P N points in P. The interpolation matrix φ is square of even size, and there are 1  2 pd 2 `3dq columns and rows; for example when d " 4, 5.If we assume P such that tdetpφpx 1 , y 1 , . . ., x δpdq , y δpdq qq ‰ 0u, then the only vector in the ta ιj u variables solving the linear system (11)  Recalling (4), the expected projective dimension of L d pPq, which is the linear system of polynomials of at most degree d with critical points at least in P P Conf pK 2 , nq, is max 1 2 pd 2 `3d ´2q ´2n, ´1( .In Section 5, we provide an alternative for studying the even dimension case in Proposition 1.
4 Essentially determined polynomials of degree three

A linear system
In order to apply elementary methods, we introduce a very simple configuration of four points, depending essentially of the fourth one px 4 , y 4 q.Secondly, we must find a polynomial f px 4 , y 4 , x, yq with a critical point set containing the above simple configuration.Let

A
.
" xypx `y ´1qpx `yqpx ´1qpy ´1q " 0 ( be an arrangement of six K-lines; it is illustrated in Fig. 1.a. a) Figure 1: a) The line arrangement A (of double lines) and the triangle " tV 1 , V 2 , V 3 u.b) The analogous objects under the linear map R, sending A to A and to ∆.It shall be convenient to write the Eq. ( 14) as a map to the space of polynomials Proof.Let the following be a polynomial f px, yq " a 1 x 3 `a2 x 2 y `a3 xy 2 `a4 y 3 `a5 x 2 `a6 xy `a7 y 2 `a8 x `a9 y P Krx, ys 0 ď3 .
1.If px 4 , y 4 q tends to be in a line , then the polynomial f px 4 , y 4 , x, yq in (17) has two lines of critical points in the respective pair of parallel K-lines L α , L β , in the arrangement tA px, yq " 0u. Figure 4 provides a sketch up to affine transformations.2. If px 4 , y 4 q tends to be the vertex p0, 0q P , then the polynomial f px 4 , y 4 , x, yq in (16) becomes f p0, 0, x, yq " 1 3 px 3 `y3 q ´px 2 y `xy 2 q ´1 2 px 2 `y2 q `xy.As is expected, the curve tf p0, 0, x, yq " 0u has a cusp of multiplicity two at (0,0), see Fig. 4. The same is valid if px 4 , y 4 q tends to be any other vertex p1, 0q, p0, 1q of .Figure 4 shows f p1, 0, x, yq, corresponding to V 2 " p0, 1q denoted as V 2 in the figure .Remark 5. Let P be any configuration of four points.Thus L 3 pPq ‰ H, i.e. there exists a non constant degree three polynomial having critical points at least in P.

Affine classification of quadrilateral configurations
We now study the independence of the previous results §4. 1, with respect to the coordinate system.
A valuable tool in the study of polynomials of degree three is the action of the group of affine automorphisms of K 2 , say Aff pK 2 q.It is a six K-dimensional Lie group.Let Aff pK 2 q acts on the space of polynomials of degree d as Aff pK 2 q ˆKrx, ys This action is rich enough and yet treatable.The affine group acts on configurations such as Aff pK 2 q ˆConf pK 2 , nq ÝÑ Conf pK 2 , nq, pT, Pq Þ ÝÑ T ´1pP q. (21) Thus, if f P Krx, ys "d has n isolated critical points, say P P Conf pK 2 , nq, then f ˝T has critical points at T ´1pP q.Hence, a useful associated object is the quotient space of quadrilateral configurations up to affine transformations.
A key point is the number of affine maps tT j u, depending on P 0 to be computed in Corollary 2.
In accordance with Fig. 1 and 3, the triangles , ∆ determine the points, line arrangements and regions below.
) sketched as six double lines.A was already described in the introduction and in (13).
), sketched as six blue lines, where N 1 , N 2 , N 3 are the axis of symmetry of .The lines N 1 , N 2 , N 3 are fixed under σ 1 , σ 2 , σ 3 in Aff pR 2 q leaving invariant .The lines N 4 , N 5 , N 6 determine the triangle C 1 , C 2 , C 3 .Naturally these points and arrangements are in correspondence under the map R in (24).
‚ In case K " R, we have two open connected regions in R 2 ; convex quadrilateral configurations when px 4 , y 4 q P Q 1 (aquamarine) and non convex for Q 2 (magenta).Analogously, we have Q 1 " RpQ 1 q and Q 2 " RpQ 2 q.Moreover, the boundary of Q 1 , Q 2 shall be described by using the isotropy of the respective configurations.
Lemma 2. Let P P Q be a generic quadrilateral configuration in K 2 as in (22).If the affine isotropy group of P Aff pK 2 q P ." tT P Aff pK 2 q | T ´1pP q " Pu is non trivial, then it is isomorphic to one of the subgroups below.
Case 1. Aff pK 2 q P -Symp3q if and only if up to affine transformation P has vertices in an equilateral triangle and its center.
Case 2. Aff pK 2 q P -Z 2 ˆZ4 if and only if up to affine transformation P is a rhombus (its vertices determine a pair of two parallel lines).
Case 3. Aff pK 2 q P -Z 2 if and only if up to affine transformation i) P " tp0, 0q, p1, 0q, p1{2, ?3{2q, px 4 , y 4 qu where px 4 , y 4 q is a fixed point under the reflection σ 1 2 with axis N 2 in the isotropy of the triangle ∆ and it is different of the center of ∆, or ii) P is a trapezoid, its vertices determine two parallel lines, different from a rhombus.l Corollary 2. Let P 0 be a generic quadrilateral configuration, the following assertions are equivalent.1) P 0 has a trivial isotropy group Aff pK 2 q P0 " id.
We note that gpV j , q are non affine maps.
Proof.The choice of one vertex V j P ∆, determines an opposite side ∆.Without loss of generality, we consider the vertex V 2 " p1, 0q P ∆ and L 1 " ty ´?3x " 0u Ă A is the opposite side; see Fig. 2. For fixed j " 2, we consider V 4 .Let L be the line by V 4 and V 2 ; L is the red line in Fig. 2. We assume that L 1 and L are non parallel.There exists a unique K-affine embedding j : Secondly, we shall extend this definition for V 4 P K 2 zL 1 .In order to avoid cumbersome computations, the coordinates tpx, yqu in (24) are more suitable.Assume P " tp0, 0q, p1, 0q, p0, 1q, px 4 , y 4 qu, the vertex is V 2 " p1, 0q P and L 1 " tx 4 " 0u is the opposite side.The analogous definition provides the rational map gpV 2 , q : K 2 ztx 4 px 4 ´1q " 0u ÝÑ K 2 ztx 4 px 4 ´1q " 0u, It enjoys the properties described below.
‚ gpV 2 , q is a birational map of K 2 .‚ g ´1pV 2 , q " gpV 2 , q, i.e. it is an involution.‚ The point V 2 and the line tx " ´1u are fixed under gpV 2 , q.
‚ The poles of the map gpV 2 , q are localized at tx " 0u and tx ´1 " 0uztp0, 1qu.Thus, strictly speaking the map is a K-analytic diffeomorphism on K 2 ztxpx ´1q " 0u.In the synthetic definition (26), L 1 and L are non parallel.This originates the pole of gpV 2 , q at tx ´1 " 0u.
‚ A straightforward computations shows that the line arrangements A and B (double and blue lines in Fig. 3) are poles or remain invariants under g 2 pV 2 , q. Summarizing, we define (26) as gpV 2 , q ." R ˝gpV 2 , q ˝R´1 .Finally, given V 4 and gpV 2 , V 4 q, there exists a unique transformation T P Aff pK 2 q, which leaves the line L 1 fixed so that T pV 4 q " gpV 2 , V 4 q; see Fig. 3.Under T , the quadrilateral configurations tp0, 0q, p1, 0q, p1{2, ?
The point gpV 2 , V 4 q determines an affine map T between generic quadrilateral configurations.
The other vertices of the triangle ∆ determine rational maps gpV 1 , q, gpV 3 , q, both enjoy analogous properties.Remark 6.Three blue lines in Fig. 3 correspond to the fixed points under the reflection symmetries Symp3q of ∆.By using (26), the complete configuration of six blue lines N 1 , . . ., N 6 is invariant under the three transformations gpV j , q.We leave this assertion for the reader.
2. For K " C, the quotient Q is a connected complex surface.3.For K " R, the quotient has two connected components Q " Q 1 Y Q 2 and singular points with local models Some comments are in order.Figure 3 illustrates the fundamental domains for π over K " R. The double lines A " L 1 Y . . .Y L 6 in Fig. 1-4 correspond to forbidden positions for px 4 , y 4 q.Moreover, px 4 , y 4 q P Q 1 means a non convex quadrilateral configuration; px 4 , y 4 q P Q 2 determines a strictly convex quadrilateral configuration.
Proof.The set theoretical construction of the quotient is simple, and we describe its projection π in (28).Given P 0 P Q, we apply an affine transformation R ˝Tj in (24) sending it to R ˝Tj pPq " tp0, 0q, p1, 0q, p1{2, ?3{2q, V 4 " px 4 , y 4 qu.Case 1.The isotropy is trivial Aff pK 2 q P " id.There are exactly 24 different choices for R ˝Tj , as in Lemma 2; we have that π has as target K 2 " tpx 4 , y 4 qu.
In order to describe its analytic properties, recall that the Klein four-group K is isomorphic to Z 2 ˆZ2 .It is such that each element is self-inverse (composing it with itself produces the identity) and composing any two of the three non-identity elements produces the third one; see [2] p. 87.Moreover, the group Symp4q is of order 24, having a Klein four-group K as a proper normal subgroup; thus Symp3q " Symp4q{K.We recognize K " tid, gpV j , q | j P 1, 2, 3u The plane R 2 zA with coordinates tx 4 , y 4 u parametrizes the quadrilateral configurations a fundamental domain for the moduli space of quadrilateral configurations, up to Aff pK 2 q-equivalence.There are 24 copies of the fundamental region Q.We colored Q 2 and its copies pink or blue (resp.Q 1 and its copies aquamarine or magenta) tiles for strictly convex (resp.non convex) quadrilateral configurations.
as the group in Lemma 3. Recall (23) and consider the homomorphism given by ϕ : Symp3q ÝÑ AutpKq, σ Þ ÝÑ σ ´1 α ˝gpV j , q ˝σα px 4 , y 4 q.The semidirect product of K and Symp3q determined by ϕ is Symp4q " K ¸ϕ Symp3q, see [2] p. 133.Hence we have a representation of Symp4q in the birational transformations of K 2 zA and is the quotient space.See [16] for a general theory of the quotients of complex manifolds under a discontinuous group of automorphisms.Assertion (1) is done.For assertion (2), K " C; note that K 2 zA is a connected complex manifold.The local behavior of this complex quotient at the points with non trivial isotropy Z 2 at the lines N 1 , N 2 , N 3 is known to be non-singular (because of C. Chevalley [7], see also [12]).For C the isotropy is Symp3q and the same references describe the local structure of the quotient.
For assertion (3), K " R; clearly the convexity or non convexity of a quadrilateral configurations are affine invariants, whence there are two connected components.At the points C, . . ., C 4 and lines N 1 , N 2 , N 3 where the isotropy of the quadrilateral configurations is non trivial, the quotient (29) has singularities; it is an orbifold.As final step in the proof of Theorem 1, we consider the action on projective classes A : Aff pK 2 q ˆP rojpKrx, ys "3 q ÝÑ P rojpKrx, ys "3 q, pT, rf sq Þ ÝÑ rf ˝T s. (30) This action provides an Aff pK 2 q-bundle structure on Krx, ys "3 .Denote the stabilizer or isotropy group of rf s P P rojpKrx, ys "3 q by Aff pK 2 q rf s ." tT P Aff pK 2 q | f ˝T " λf, λ P K ˚u.Equations ( 15) and (24) provide bijective correspondence between generic quadrilateral configuration in px 4 , y 4 q P K 2 zA and projective classes of polynomials rf pR ´1px 4 , y 4 q, x, yqs.If P P Q, then we verify that the isotropy of the quadrilateral configuration Aff pK 2 q P is isomorphic to Aff pK 2 q rf s .Thus, we have a section f ˝R´1 : K 2 ztAu ÝÑ P rojpKrx, ys "3 q, px 4 , y 4 q Þ ÝÑ rf pR ´1px 4 , y 4 q, x, yqs and a diagram P rojpKrx, ys "3 q c π rf pR ´1px 4 , y 4 q, x, yqs here π is the projection of classes from the action (30).The Aff pKq-orbit of a projective class rf s P Krx, ys "3 is homeomorphic to Aff pK 2 q{Aff pK 2 q rf s .Obviously, Krx, ys "3,id is open and dense in Krx, ys "3 .The proof of assertion 1, Theorem 1 is done.
Remark 8.For K " R, the fundamental domain Q 1 Y Q 2 determines the bifurcation diagram of the respective Hamiltonian vector fields, see Fig. 4. By construction, Q 1 has two boundaries and one vertex C and Q 2 has one boundary (without extreme points).
We summarize the results in Table 2.
Proof.In assertion (1), up to an affine transformation we can assume y 4 " 1.The corresponding cubic polynomial takes the form f px, yq " a 4 `2y 3 ´3y 2 ˘, where a 4 P K ˚.
For assertion 2, we search for polynomials f px, yq P Krx, ys 0 ď3 with at least four affine collinear critical points.The matrix of Eq. ( 17) results in the cubic polynomials f px, yq " a 3 xy 2 `a4 y 3 `a7 y 2 " y 2 pa 3 x `a4 y `a7 q, ra 3 , a 4 , a 7 s P KP 2 , with a line of critical points in ty " 0u.
Example 2. The elementary methods provide an insight in the case of a double point in Σpf q.Let P 2 " tp0, 0q, p1, 0q, p0, 1q, p0, 0qu be such a configuration.A basis for L 3 pP 2 q is x 3 ´3x 2 , y 3 ´3y 2 , x 2 y `xy 2 ´xy.The first and second polynomials have lines of singularities, the third one four isolated critical points.The family of polynomials is f pa 1 , a 2 , a 4 , x, yq " a 1 px 3 ´3x 2 q `a2 px 2 y `xy 2 ´xyq `a4 py 3 ´3y 2 q, ra 1 , a 2 , a 4 s P KP 2 .
As is expected, for values pa 1 , a 2 , a 4 " a 2 2 {9a 1 ( the two dimensional family f pa 1 , a 2 , a 4 , x, yq determines polynomials with three isolated singular points, one of them of multiplicity two, see Fig. 4.
The interpolation matrix φ, Eq. (33), is square.Hence, for an open and dense set of configurations tPu Ă Conf pK 2 , 7q such that tdetpφq " 0u, the resulting space of polynomials of degree four having these P as critical points is empty.In order to overcome this situation, we introduce the following concept.
Proposition 2. Let Krx, ys 0 ďd having even dimension.1.The interpolation curve I of P 0 describes the position of the δpdq-th point such that dim K pL d pP 1 qq ě 0.
2. There exists a Zariski open set tP 0 u Ă Conf pK 2 , δpdq ´1q such that the associated tIu are algebraic curves of degree 2d ´2 in K 2 .
After fixing the configuration P 0 , the associated linear system only has free variables x, y, and the linear system is as follows ¨. . .
The determinant of this matrix has x 2d´2 as higher degree monomial, we are done.
We describe some interpolation curves I.
Example 3. Let f P Krx, ys 0 ď4 be a polynomial having degree four and let P 0 " tpx ι , y ι q | ι P 1, . . ., 6u be a fixed configuration of six different critical points of f .1.If three points of P 0 are in a line tx " 0u and two points are in tx " 1u, then the interpolation curve I, of P 0 , is given by Ipx, yq " `´1152y 2 4 y 2 5 py 4 ´1q 2 x 6 px 6 ´1q ˘xpx ´1qpx ´x6 qgpx, yq.
The I is reducible and singular, it is the product of three parallels lines and a polynomial gpx, yq that pass through the six points in P 0 .2. Let P 0 " tpx ι , y ι q | ι P 1, . . ., 6u be any configuration of six points in the grid of nine points G " txpx ´1qpx ´c1 q " 0u X typy ´1qpy ´c2 q " 0u, where c 1 , c 2 R t0, 1u.Therefore, the interpolation curve I, associated to the seventh point px 7 , y 7 q, is the product of the six lines defining G. 3. Let P " tpx ι , y ι q | ι P 1, . . ., 6u be a configuration of six critical points of f .If the six points are distributed in a conic Q, then the interpolation curve I, associated to the seventh point px 7 , y 7 q, contains the conic.That is, I " Qg for some g P Krx, ys 0 ď4 .A complete study of the interpolation curves I arising from configurations of six points is the goal of a future project.
P ci is the base locus of the pencil of curves.
Moreover, the choice of an ordered pair of polynomial functions from (37), not just curves say `aF px, yq `bGpx, yq, cF px, yq `dGpx, yq determines a SLp2, Kq-pencil of polynomial vector fields FpP ci q " " X M " ´`cF px, yq `dGpx, yq ˘B Bx ``aF px, yq `bGpx, yq ˘B By ˇˇM " p ´c ´d a b q P SLp2, Kq * .
The condition detpMq ‰ 0 is equivalent with the fact the zero locus of the vector field X M coincides with P ci .Lemma 6.Let U d Ď XpK 2 q ďd´1 be the open and dense set of polynomial vector fields of degree d ´1, having exactly pd ´1q 2 zeros in P ci Ă Conf pK 2 , pd ´1q 2 q.Assume that P ci has trivial isotropy group in Aff pK 2 q.In U d there exists an analytic SLp2, Kq-bundle structure as follows Proof of the Corollary.The family X M with a grid of pd ´1q For pa, dq ‰ pa, 0q, p0, dq, each polynomial f pa, d, x, yq P Krx, ys 0 ďd in (42) has pd ´1q 2 Morse critical points.In fact, at each point p P P a very simple observation with the Taylor series shows that f pa, d, x, yq " r ax 2 `r by 2 `O3 px, yq, where r a r b ‰ 0.

Closing remarks
Let XpK 2 q ďd´1 be the space of polynomial vector fields tXu of at most degree d´1 on K 2 .A general and natural question is as follows.Under what conditions a polynomial vector field X on K 2 is essentially determined by its configuration of zeros ZpXq in K 2 ?
In simple words, a vector field X is essentially determined (in XpK 2 q ďd´1 ) by its configuration of zeros ZpX f q, if for any Y P XpK 2 q ďd´1 satisfying ZpXq Ă ZpY q Ă K 2 , then X " λY .Recalling that for affine degree d the number of isolated singularities of the associated singular holomorphic foliation FpX q on the whole CP 2 is pd ´1q 2 `d, the hypothesis of multiplicity one must be understood for all these points.Proposition 1 confirms that in the Hamiltonian case only δpdq ď pd ´1q 2 points are required.
Recall that X. Gómez-Mont and G. Kempf, [13], established in the complex rational case the following deep result, that also enlightens the real case.
A meromorphic vector field X on CP m , m ě 2, of degree r ě 2, with critical points having all its zeros of multiplicity one is completely determined by its zero set.
Moreover, J. Artes, J. Llibre, D. Schlomiuk and N. Vulpe, [4], [5] prove the following: A polynomial vector field X on K 2 of degree two, is completely determined by the position of its seven critical points (including the points at infinity).
As far as we know, over K " C the more general result is due to A. Campillo and J. Olivares, [6]: A singular holomorphic foliation X on CP 2 of degree r ě 2, w is completely determined by its singular scheme.See C. Alcántara et al. [1] for recent developments regarding foliations with multiple points.We summarize our results as follows.
Corollary 7. A polynomial Hamiltonian vector field X f on K 2 of degree two is completely determined (in the space of polynomial vector fields of degree 2, up to a scalar factor λ P K ˚) by its zero points, when they are four isolated points different from tp0, 0q, p1, 0q, p0, 1q, p1, 1qu, up to affine transformation.
Our hope is that the explicit results in this paper can illustrate the classification of polynomials Krx, ys up to algebraic equivalence AutpK 2 q; see [11] and [18] for this order of ideas.This potential application is the subject of a future project.

Corollary 5 . 2 ¯ˇa d " 1
The one dimensional holomorphic family of Hamiltonian vector fields of the polynomials " f pa, d, x, yq "

Table 1 :
Dimensions and values for the interpolation problem.

Table 2 :
Dimension, generators and isotropy for L 3 pPq, where P is a configuration with 4 points (3 simple points and a double one in the last row).
2points is Hamiltonian if and only if M P `0 ´d a 0 ˘( -K 2 Ă SLp2, Kq.In fact, ω m " paF pxq`bGpyqqdx`pcF pxq`dGpyqqdy " 0 is exact if and only if bGpyq y " cF pxq x .The equality holds only for b " c " 0. The respective vector subspace of polynomials pPq Ą trf pa, d, x, yqsu and dim K pL d pPqq " 1.(43) d