Sign-change diminishing systems of functions of many variables (Q1332303)

Let \(y_ i\), \(z_ i\), \(i=1,\dots,r\), be real numbers. We write \[ (y_ 1,y_ 2,\dots,y_ r)\prec_ k (z_ 1,z_ 2,\dots,z_ r) \] (respectively, \((y_ 1,y_ 2,\dots,y_ r)\prec^ k(z_ 1,z_ 2,\dots, z_ r))\) iff there do not exist integers \(1\leq s_ 1\leq\cdots\leq s_ k\leq r\) such that \((-1)^{k-i}y_{s_ i}> (- 1)^{k-i} z_{s_ i}\) (respectively, \((-1)^{k-i}y_{s_ i}\geq (- 1)^{k-i} z_{s_ i})\), \(i= 1,\dots, k\). A system of functions \[ f_ 1(x)= f_ 1(x_ 1,\dots,x_ m),\dots, f_ n(x)= f_ n(x_ 1,\dots,x_ m)\tag{1} \] defined on \(M\subset \mathbb{R}^ m\) is called sign-change diminishing iff for any \(y, z\in M\), \(y\neq z\), and integer \(k\geq 1\) the relation \(y\prec_ k z\) implies \((f_ 1(y),\dots, f_ n(y))\prec^ k(f_ 1(z),\dots, f_ n(z))\). It is known that the system (1) with \[ f_ i(x)= a_{i_ 1}, x_ 1+\cdots + a_{i_ m} x_ m+ b_ i,\quad i= 1,\dots,n, \] \(M= \mathbb{R}^ m\) is sign change diminishing iff the matrix \((a_{ij})^{n,m}_{i=1,j=1}\) has all its minors positive. The nonlinear case is considered in this paper. It is proved that if \(M\) is an open parallelopiped or simplex and the matrix \(\left({\partial f_ i(x)\over \partial x_ j}\right)^{j=1,2,\dots,m}_{i= 1,\dots,n}\), \(x\in M\) has all the minors positive then the system (1) is sign change diminishing.