On zero curves of bivariate polynomials
In this paper some properties of the bivariate polynomials \[ f(x,y)= \sum^m_{i=0} \sum^n_{j=0} a_{ij}x^iy^j; \qquad (x,y)\in\Omega= (0,\infty)\times (0,\infty) \] are investigated. Two conditions on the signs of the coefficients of a bivariate polynomial which ensure that the zero set is a single curve are derived. The first condition demands that all but one of the coefficients has the same sign. The second requires that the signs are ``split'' by any stright line. Both properties are demonstrated by generalizing the set of isoparametric lines to a certain two parameter family of curves. Equivalent curves are found for power, tensor-product Bernstein, exponential and triangular Bernstein polynomials. The second property will allow greater freedom when using algebraic curves for geometric modeling. The author presents the following observations: (a) If only one negative coefficient \(a_{ij}\) is an internal one \((1\leq i\leq m-1\); \(1\leq j\leq n-1)\) and if all remaining coefficients are positive then the zero set \(f_0\) is one of the following (i) empty, (ii) a single point, (iii) a single closed curve. (b) If many coefficients are zero, then the zero set \(f_0\) can have two connected components. For example the zeros of \(f(x,y)= x-3xy+2xy^2\) are the lines \(y=1\) and \(y=2\).
scientific article; zbMATH DE number 947617