In this paper, we study sums of translates on the real axis. These functions generalize logarithms of weighted algebraic polynomials. Namely, we are dealing with functions \((F\text{y},t) := J(t) + \sum _{j=1}^n K_j(t-y_j), \quad \text{y} := (y1,\ldots,y_n), \ y_1 \le \cdots \le y_n, \ t \in \mathbb{R},\) where the kernels \(K_1,\ldots,K_n\) are concave on \((-\infty,0)\) and on \((0,\infty)\) , having a singularity at \(0\) , and \(J\colon \mathbb{R}\to \mathbb{R}\cup\{-\infty\}\) isthe field function. We consider "local maxima" \(m_0(\text{y}) := \sup _{t \in (-\infty, y_1]} F(\text{y}, t), \quad m_n(\text{y}) := \sup _{t \in [y_n, \infty)} F(\text{y}, t),\) \(m_j(\text{y}) := \sup _{t \in [y_j, y_{j+1}]} F(\text{y}, t), \quad j = 1,\ldots,n-1, \) and the difference function \((D\text{y}) := (m_1\text({y})-m_0\text({y}), m_2\text({y})-m_1\text({y}), \cdots, m_n\text({y})-m_{n-1}\text({y})). \) We prove that, under certain assumptions on the kernels and the field, \(D\) is a homeomorphism between its domain and \(\mathbb{R}^n\) .
