We present a family of flattening methods of tensors which we call Kronecker-Koszul flattenings, generalizing the famous Koszul flattenings and further equations of secant varieties studied among others by Landsberg, Manivel, Ottaviani and Strassen. We establish new border rank criteria given by vanishing of minors of Kronecker-Koszul flattenings. We obtain the first explicit polynomial equations -- tangency flattenings -- vanishing on secant varieties of Segre variety, but not vanishing on cactus varieties. Additionally, our polynomials have simple determinantal expressions. As another application, we provide a new, computer-free proof that the border rank of the $2\times2$ matrix multiplication tensor is $7$.

翻译：我们提出了一族称为Kronecker-Koszul展平的张量展平方法，该方法推广了著名的Koszul展平以及由Landsberg、Manivel、Ottaviani和Strassen等学者研究的割线簇的进一步方程。我们建立了由Kronecker-Koszul展平子式消失给出的新边界秩判据。我们获得了首批显式多项式方程——切性展平方程——这些方程在Segre簇的割线簇上消失，但在仙人掌簇上不消失。此外，我们的多项式具有简洁的行列式表达式。作为另一应用，我们提供了一种无需计算机辅助的新证明，表明$2\times2$矩阵乘法张量的边界秩为$7$。