A novel necessary and sufficient condition for the positivity of a binary quartic form

In this paper, by considering the common points of two conics instead of the roots of the binary quartic form, we propose a novel necessary and sufficient condition for the positivity of a binary quartic form using the theory of the pencil of conics. First, we show the degenerate members of the pencil of conics according to the distinct natures of the common points of two base conics. Then, the inequalities about the parameters of the degenerate members are obtained according to the properties of the degenerate conics. Last, from the inequalities we derive a novel criterion for determining the positivity of a binary quartic form without the discriminant.

The problem of determining the global positivity or nonnegativity of forms has many applications. For instance, in the study of two-dimensional nonlinear systems and other mathematical and physical fields, we often need to deal with the positive definite or negative definite problems of a binary quartic form [1,2]. So far, about the positive definite problem of a binary quartic form, there has existed some literatures. For example, Chin [3] used the canonical forms of a real quartic equation to propose a necessary and sufficient condition for no real roots. Gadenz and Li [4] published a systematic method for determining the positive definiteness of the binary quartic form. Their method adopted the testing of the permanence and variations of sign in the sequence of principal minors of the corresponding Hankel matrix. Jury and Mansour [5] used Ferrari's solution to obtain condition for positivity for a quartic equation. In the same year, Fuller [6] also gave similar criteria in terms of bigradient determinants. Ku [7] gave a criterion for the positive definiteness of the general quartic form from the known solution. However, his result was not complete. Wang  (2) be a binary quartic form on real number field ℝ. In terms of the coefficients of (2), the following quantities can be defined [7]: 1) ∆= 0, = 0, 12 2 − 0 2 = 0, > 0; 2) ∆> 0, ≥ 0; 3) ∆> 0, < 0, 12 2 − 0 2 < 0.
Compared with the criterion in [4], this criterion based on the discriminant of a quartic polynomial is superior in that the conditions contained in this criterion are stated in the form of a set of inequalities which are explicit functions of the coefficients of the quartic form. Besides for this common criterion, Hasan et al. [9] presented an alternative criterion. They made use of some of the results of the quadratic programming theory to show that testing for positive semi-definiteness of the quartic form (1) was reduced to a test whether there is a real number such that the parametric matrix If yes, the criterion in [9] will be simplified. In this paper, we will see that the answer to this question is yes. We use the relationship between the common points of two conics and the degenerate members of the pencil of conics to address this question and further derive a novel criterion for determining the positivity of a binary quartic form without the discriminant. We show that given a monic binary quartic form (1), the following quantities can be defined:

Notations and Preliminaries
Throughout this paper, let 1 and 2 denote respectively two real symmetric matrices as follows: , and let the matrix be where is a variable real parameter. Since every 3 × 3 real symmetric matrix denotes a conic in the complex projective plane, is a pencil of conics determined by the base conics 1 , 2 .
Obviously, using the matrix , the binary quartic form (1) can be represented as follows: ].
On the other hand, for arbitrary , not all zero, [ 2 , , 2 ] is the projective coordinate of the point of the conic 2 , so we can use the relative position of the conics 1 , 2 to address the determination of positivity of the binary quartic form (1). We assert that the binary quartic form (1)  It is well known that the conics 1 , 2 have four common points (which need not all be distinct). Further the relative position of the conics 1 , 2 has nine cases according to the nature (generic or non-generic, real or complex) of their common points [10,11]. They are presented in Table 1. Now, we discuss the degenerate members of the pencil since they are closely related to the relative position of the conics 1 , 2 . The pencil contains three degenerate members (which need not all be distinct) [12]. Their parameters are the roots of the cubic equation det ( ) = 0. The degenerate conics are the line-pairs through four common points of 1 and 2 . So, according to the nature of the common point in Table 1, the degenerate member in the pencil may concretely consist of one of as follows: (1) a real line-pair (two real lines); (2) a complex line-pair (two complex lines); (3) a complex conjugate line-pair (two complex conjugate lines); (4) a real repeated line.   Table 1 b 2 , 3 are complex conjugate The middle column of Table 2 gives the concrete type of every degenerate member in nine cases. The different type of degenerate member can provide important information about its parameter . More formally, we have the following lemmas. From the above three lemmas, it is easy to obtain the relation between 3 2 /4 and the parameters 1 , 2 , 3 of the three degenerate members of the pencil in every case that is given in the middle column of Table 2. The results are presented in the last column of Table  2. It is worth noting that this is a one-to-one correspondence between the middle column of Table 2 and the last column of Table 2.
For the other eight cases in Table 2, 0 must be real and furthermore the relation between 0 and 3 2 /4 and the relation between ( 0 ) and 0 can be obtained easily. The results are presented in the last column of Table 3. Table 3 Properties of 0 and ( 0 ) Case a Property of 1 , 2 , 3 Property of 0 and ( 0 ) , ( 0 ) = 0 a Corresponding to Table 1 and Table 2 3

. The Main Results
From the foregoing discussion, we know that the binary quartic form (1) is positive semidefinite if and only if there is no real simple intersection point between the conic 1 and the conic 2 , i.e., Case 2, Case 5, Case 6, Case 7 and Case 9 in Table 1. In particular, the form (1) is positive definite if and only if there is no real common point between the conic 1 and the conic 2 , i.e., Case 2, Case 7. Consequently, we can use the properties of 0 and ( 0 ) in Table 3 to obtain a simplified necessary and sufficient condition compared with [9] for determining the positivity of the form (1).

Theorem 1. Given a monic binary quartic form
( , ) = 4 + 3 3 + 2 2 2 + 1 3 + 0 4 . Thus, if the form (3) is positive semi-definite, then there is no real simple intersection point between two base conics, i.e., Case 2, Case 5, Case 6, Case 7 and Case 9 in Table 1. Further, from Table 3, we have 0 ≥ 3 2 /4 and ( 0 ) ≥ 0, namely the matrix 0 is positive semidefinite. In particular, if the form (3) is positive definite, then there is no real common point between two base conics, i.e., Case 2, Case 7. Further, from Table 3, we have 0 > 3 2 /4 and ( 0 ) > 0. Consequently, according to Sylvester's criterion, the matrix 0 is positive definite. This completes the proof of the theorem. □ From the proof of Theorem 1, we can obtain directly a novel criterion for determining the positivity of a monic binary quartic form. It is well known that the negativity of − 4 + 3 3 + 2 2 2 + 1 3 + 0 4 is equivalent to the positivity of 4 − 3 3 − 2 2 2 − 1 3 − 0 4 . Consequently, according to the obtained conclusions, we may obtain directly the following corollaries about the negativity of a binary quartic form. is a negative semi-definite (negative definite) matrix.