Abstract
We prove that the zero set of a 4-nomial in n variables in the positive orthant has at most three connected components. This bound, which does not depend on the degree of the polynomial, not only improves the best previously known bound (which was 10) but is optimal as well. In the general case we prove that the number of connected components of the zero set of an m-nomial in n variables in the positive orthant is lower than or equal to (n+1)m-121 + (m - 1)(m - 2)/2, improving slightly the known bounds. Finally, we show that for generic exponents, the number of non-compact connected components of the zero set of a 5-nomial in three variables in the positive octant is at most 12. This strongly improves the best previously known bound, which was 10,384. All the bounds obtained in this paper continue to hold for real exponents.
Highlights
Descartes’ Rule of Signs provides a bound for the number of positive roots of a given real univariate polynomial which depends on the number of sign changes among its coefficients but not on its degree
One of its consequences is that the number of positive roots of a polynomial with m monomials is bounded above by m − 1
Let us prove that each non-compact connected component of W intersects at least one of the sets S1, . . . , Sn, T1, . . . , Tn
Summary
Descartes’ Rule of Signs provides a bound for the number of positive roots of a given real univariate polynomial which depends on the number of sign changes among its coefficients but not on its degree. One of its consequences is that the number of positive roots of a polynomial with m monomials is bounded above by m − 1. Many attempts have been made to generalize Descartes’ Rule of Signs (or its corollaries) to a larger class of functions. Even though this task has not yet been completed, important advances have been made [2]–[4], [8]. N denotes the set of positive integers.
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