Is the Signed Zarankiewicz Number the Same as the Recursive-line Zarankiewicz Number?

Abstract: Löfberg and Qi introduced the second order Zarankiewicz number (z_2), the recursive-line Zarankiewicz number (z_{RL}), and the signed Zarankiewicz number (z_{SL}) for doubly simple biquadratic forms. It was shown that [ z_2(m,n)\ge z_{SL}(m,n)\ge z_{RL}(m,n) ] for all (m) and (n). However, there was no evidence that there exist particular (m) and (n) such that (z_{SL}(m,n)>z_{RL}(m,n)). The motivation for introducing (z_{SL}) was as follows: during the study of the exceptional case (m=15), (n=6), Löfberg and Qi showed that [ z_2(15,6)=z_{SL}(15,6)=60, ] but the exact value of (z_{RL}(15,6)) was unknown then. In this paper we show that [ z_{RL}(15,6)=60. ] This eliminates the motivation for introducing (z_{SL}). Whether (z_{SL}(m,n)=z_{RL}(m,n)) in general remains an open problem. Recently, Lebedev presented an explicit construction separating the augmented Zarankiewicz number (z_A) from the limited augmented Zarankiewicz number (z_L) at (m=n=1893). We hope that the separation problem for (z_{SL}) and (z_{RL}) can also be solved. We also present the exact values of (z_{RL}(m,6)) for (6\le m\le 16).