Discriminantly separable polynomials and quad-equations

We classify the discriminantly separable polynomials of degree two in each of three variables, defined by a property that all the discriminants as polynomials of two variables are factorized as products of two polynomials of one variable each. Our classification is based on the study of structures of zeros of a polynomial component $P$ of a discriminant. This classification is related to the classification of pencils of conics in a delicate way. We establish a relationship between our classification and the classification of integrable quad-equations which has been suggested recently by Adler, Bobenko, and Suris.


Contents
1 Introduction: an overview on discriminantly separable polynomials 2 2 Classification of strongly discriminantly separable polynomials of degree two in three variables 5 3 From discriminantly separable polynomials to integrable quadgraphs 11 1 Introduction: an overview on discriminantly separable polynomials In a recent paper [13] of one of the authors of the present paper, a new approach to the Kowalevski integration procedure from [19] has been suggested. It has been based on a new notion introduced therein of the discriminantly separable polynomials. A family of such polynomials has been constructed there as pencil equations from the theory of conics F(w, x 1 , x 2 ) = 0, where w, x 1 , x 2 are the pencil parameter and the Darboux coordinates respectively. (For classical applications of the Darboux coordinates see Darboux's book [11], for modern applications see the book [15] and [12].) The key algebraic property of the pencil equation, as quadratic equation in each of three variables w, x 1 , x 2 is: all three of its discriminants are expressed as products of two polynomials in one variable each: where J, P are polynomials of degree 3 and 4 respectively, and the elliptic curves Γ 1 : y 2 = P (x), Γ 2 : y 2 = J(s) are isomorphic (see Proposition 1 of [13]) .
There are two natural and important questions in this context: 1) Are there any other discriminantly separable polynomials of degree two in each of three variables, beside those constructed from pencils of conics? In addition, the question is to perform a classification of such polynomials.
2) Are there other integrable dynamical systems related to discriminantly separable polynomials?
The main issue of this paper is to address these two key questions.
In order to make precise the first question, one needs to specify a gauge group or the classes of equivalence up to which a classification would be performed. This leads to the group of Möbius transformations, as introduced in Corollary 3 of [13]: The family of discriminantly separable polynomials in three variables of degree two in each of them, constructed from pencils of conics served as a motivation to introduce more general classes of discriminantly separable polynomials. Let us recall here the definitions from [13]: a polynomial F (x 1 , . . . , x n ) is discriminantly separable if there exist polynomials f i (x i ) such that for every i = 1, . . . , n . The classification of strongly discriminantly separable polynomials F(x 1 , x 2 , x 3 ) of degree two in each of three variables modulo fractional-linear transformations from formulae (3) as the gauge group, is one of the main tasks of the present paper.
This classification is heavily based on the classification of pencils of conics (see [15], for example, for more details about pencils of conics). In the case of general position, the conics of a pencil intersect in four distinct points, and we code such situation with (1, 1, 1, 1), see Fig. 1, and denote it by A. It corresponds to the case where polynomial P has four simple zeros. In this case, the family of strongly discriminantly separable polynomials coincides with the family constructed in [13] from a general pencil of conics. This family, as it has been indicated in [13], corresponds to the two-valued Buchstaber-Novikov group associated with a cubic curve Γ 2 : y 2 = J(s).
Two-valued groups of the form (Γ 2 , Z 2 ) have been introduced in [7]. The socalled Kowalevski change of variables appeared to be an infinitesimal of the two-valued operation in this group, see [13]. (The theory of n valued-groups originates from a pioneering paper [9]. For a modern account see [6], and for higher-genus analogues, see [8].) Other cases within the classification with a nonzero polynomial P correspond to the situation where: B the polynomial P has two simple zeros and one double zero, we code it (1, 1, 2), and the conics of the corresponding pencil intersect in two simple points, and they have a common tangent in the third point of intersection, see Fig. 2; C the polynomial P has two double zeros, code (2,2), and the conics of the corresponding pencil intersect in two points, having a common tangent in each of the points of intersection, see Fig. 3; D the polynomial P has one simple zero and one triple zero, (1,3), and the conics of the corresponding pencil intersect in one simple point, and they have another common point of tangency of third order, see For more details, see Section 2.
Referring to the second key question, about constructing other integrable dynamical systems related to discriminantly separable polynomials, we address it in the following direction: to trace a connection between the discriminantly separable polynomials of degree two in each of three variables with the discrete integrable systems on quad-graphs. See [3] and [4] for more detail about the quad-graphs; some basic notions from there are collected in Section 3. For each discriminantly separable polynomial we construct explicit discrete integrable system in Section 3. Thus, we establish a natural correspondence between our classification of discriminantly separable polynomials of degree two in three variables, mentioned above and described in Section 2, and the classification of integrable quad-graphs performed in [4]. This correspondence gives, as a fit back benefit, a geometric interpretation of constructions of Adler, Bobenko and Suris in terms of pencils of conics. Developing further this correspondence, in the case coded (1, 1, 1, 1), with a polynomial P with four distinct zeros, we come to the conclusion that such integrable quad-graphs are related to the two-valued Buchstaber-Novikov groups associated with elliptic curves, i.e. those defined on CP 1 .
Recently, a new class of geometric quad-graphs, of the so-called line congruences, associated with pencils of quadrics has been introduced in [16].

Classification of strongly discriminantly separable polynomials of degree two in three variables
In this section we will classify strongly discriminantly separable polyno- which are of degree two in each variable, modulo a gauge group of the following fractional-linear transformations Let be a strongly discriminantly separable polynomial with Here, by D x i F(x j , x k ) we denote the discriminant of F considered as a quadratic polynomial in x i .
By plugging (5) into (6) for a given polynomial P (x) = Ax 4 +Bx 3 +Cx 2 + Dx + E, we get a system of 75 equations of degree two with 27 unknowns a ijk .

Theorem 1
The strongly discriminantly separable polynomials F(x 1 , x 2 , x 3 ) satisfying (6) modulo fractional linear transformations are exhausted by the following list depending on the structure of the roots of a non-zero polynomial P (x) : (A) P has four simple zeros, with the canonical form P (x) = (k 2 x 2 −1)(x 2 − 1), (B) P has two simple zeros and one double zero, with the canonical form P (x) = x 2 − e 2 , e = 0, (D) P has one simple and one triple zero, with the canonical form P (x) = x,   Proof. Proof is done by a straightforward calculation by solving the system of equations (6) for the canonical representatives of the polynomials P. For example, in the case when as the solutions of the system (6) we get the polynomials: The polynomials F A1 1,2 and F A2 1,2 are equivalent modulo the fractionallinear transformations (4) x i → −x i , i = 1, 2, 3. The same is true for the polynomials F A3 1,2 and F A4 1,2 .
Further, acting by the transformations we get the gauge equivalence between F A1 1,2 and F A3 1,2 .
In the same way we get the polynomials in the cases [B]-[D].
We consider more closely the case when P (x) has four simple zeros.
Proposition 1 In the case when the polynomial P (x) has four simple zeros, all strongly discriminantly separable polynomials F(x 1 , x 2 , x 3 ) satisfying (6) are, modulo the fractionally-linearly transformations, equivalent to Proof. The gauge transformation Here, g 2 and g 3 denote the coefficients of the Weierstrass normal form 4x 3 − g 2 x − g 3 . For a polynomial P (x) = A + 4Bx + 6Cx 2 + 4Dx 3 + Ex 4 , these coefficients are given by Notice that expanding (7) as a polynomial of x 3 , it becomes Comparing with [7] we get Corollary 1 If a polynomial P has four simple zeros, all strongly discriminantly separable polynomials F satisfying (6) are equivalent to the two-valued group 3 From discriminantly separable polynomials to integrable quad-graphs Now, we will establish a connection between the discriminantly separable polynomials and the theory of integrable systems on quad graphs. Latter has been developed by Adler, Bobenko, Suris [3], [4], see also [1], [2]. Moreover, we are going to construct an integrable quad graph associated to an arbitrary given discriminantly separable polynomial.
Recall that the basic building blocks of systems on the quad-graphs are the equations on quadrilaterals of the form where Q is a polynomial of degree one in each argument, i.e. Q is a multiaffine polynomial. The field variables x i are assigned to four vertices of a quadrilateral as in a Figure 6. Besides depending on the variables x 1 , ..., x 4 ∈ C, the polynomial Q still depends on two parameters α, β ∈ C that are assigned to the edges of a quadrilateral. It is assumed that opposite edges carrie the same parameter. Equations of type (9) are called the quad-equations. The equation (9) can be solved for each variable, and the solution is a rational function of the other three variables. A solution (x 1 , x 2 , x 3 , x 4 ) of equation (9) is singular with respect to x i if it also satisfies the equation Q x i (x 1 , x 2 , x 3 , x 4 ) = 0.
Following [4] we consider the idea of integrability as a consistency, see Figure 7. We assign six quad-equations to the faces of the coordinate cube. The system is said to be 3D-consistent if the three values for x 123 obtained from the equations on the right, back and top faces coincide for arbitrary initial data x, x 1 , x 2 , x 3 .

Then, by applying the discriminant-like operators introduced in [4]
one can make descent from the faces to the edges and then to the vertices of the cube: from a multiaffine polynomial Q(x 1 , x 2 , x 3 , x 4 ) to a biquadratic polynomial h(x i , x j ) := δ x k ,x l (Q(x i , x j , x k , x l )) and further to a polynomial P (x i ) = δ x j (h(x i , x j )) of degree up to four. By using of relative invariants of polynomials under fractional linear transformations, the formulae that express Q through the biquadratic polynomials of three edges, were derived in [4]: A biquadratic polynomial h(x, y) is said to be nondegenerate if no polynomial in its equivalence class with respect to the fractional linear transformations, is divisible by a factor of the form x − c or y − c, with c = const. A multiaffine function Q(x 1 , x 2 , x 3 , x 4 ) is said to be of type Q if all four of its accompanying biquadratic polynomials h jk are nondegenerate. Otherwise, it is of type H. Previous notions were introduced in [4].

Lemma 1 Given a biquadratic polynomial
satisfying the condition: where the discriminant δ x i (h) is given by (10) and P (x i ) is a nonzero polynomial of degree up to four in the canonical form. Then, up to the fractional linear transformations, h is equivalent to: Proof. Proof is done by a straightforward calculation by solving the system of equations (13).
We can notice that the families of polynomials h of two variables x 1 , x 2 obtained in the previous Lemma depend on the additional parameter. For example, in the case (A), they depend on the additional free parameter h 20 . Correlation with strongly discriminantly separable polynomials F(x 1 , x 2 , x 3 ) is achived through the additional assumption that coefficients of polynomials h ij are quadratic functions of the parameter α. Now, we start with an arbitrary strongly discriminantly separable polynomial of degree two in each of three variables. We are going to construct an integrable quad-graph which corresponds to F(x 1 , x 2 , α). In order to define such an integrable quad-graph we have to provide corresponding biquadratic polynomial h = h(x 1 , x 2 ) and a multiaffine polynomial Q = Q(x 1 , x 2 , x 3 , x 4 ). The requirement that the discriminants of h(x 1 , x 2 ) do not depend on α, see [3], [4], will be satisfied if as a biquadratic polynomials h(x 1 , x 2 ) we takê
The final problem of the reconstruction of the quad-equations of type Q which correspond to the strongly discriminantly separable polynomials obtained in Theorem 1 is done with a use of the formulae (11) and with the polynomials h ij replaced byĥ ij .

Theorem 2
The quad-equations of type Q that correspond to the biquadratic polynomialsĥ are given in the following list: Proof. From (11) we get that when Recall here that P A (x) = (k 2 x 2 − 1)(x 2 − 1). Choosing In the same way for the other choicesĥ ij =ĥ I (x i , x j ), I = B, C, D, E from (11) we get the expressions forQ B -Q E .

Remark 1
The list of biquadratic polynomials from Lemma 1 contains the list of h's obtained in [4] as particular cases. More precisely, by plugging Next, with a choice Finally, for The expressions for h in [4] for P (x) = x 2 and P (x) = 1 are the same as the corresponding h in Lemma 1.
In the same manner, the list obtained in Theorem 2 contains the list of the multiaffine equations of type Q obtained in [4]: In order to see the correspondence betweenQ A = 0 and (23) we need to go back to the system 6. In the process of solving the systems under the current assumptions, a step before the final solution, one comes to the following relation: Then, (23) is obtained from (27) with α → sn(α), P A (α) → sn ′ (α) and similarly for β. If we replace the parameters α, β in (23) by then the correspondence ofQ A and (27) is obvious. The correspondence ofQ B and (24) with δ = e 2 is achieved with a change Notice here that with the last change we get and similarly for β.
The equation Q(x 1 , x 2 , x 3 , x 4 ) = 0 that corresponds to the polynomial F given with (7) is obtained in [3]: where the coefficients are a 0 = a + b, a 1 = −βa − αb, a 2 = β 2 a + α 2 b, and Previous calculations establish a relation between strongly discriminantly separable polynomials in three variables of degree two in each and biquadratic polynomials from the theory of quad-graphs which correspond to the diagonal elements of table on page 11, paper [4]. Remaining three off-diagonal cases of that table from [4], p.11, appear to be connected with the symmetrically discriminantly separable polynomials (see [13] and the Introduction for the definition of the symmetrically discriminantly separable polynomials). The next Remark is devoted to that case.
For the case h 2 , let us consider the next polynomial: Finally, for the case h 3 we take: Let us underline that the correspondences established in the current Section, provide a two way-link between the strongly discriminantly separable polynomials and the quad-graphs. As a benefit, we get a geometric interpretation of the biquadratic polynomials h from the theory of quadgraphs by relating them to the pencils of conics. (The pencils of conics appeared in the classification of the Yang-Baxter maps in [5].) Moreover, in the case [A], coded (1, 1, 1, 1), when a polynomial P has four distinct zeros, the biquadratic polynomials of quad-graphs correspond to the twovalued Buchstaber-Novikov groups associated with elliptic curves, i.e. those defined on CP 1 .