Norms of Roots of Trinomials
نویسندگان
چکیده
(a) f has two distinct roots with identical norm v if and only if p is located on a real double point of the hypotrochoid, (b) f has a root of multiplicity 2 with norm v if and only if the corresponding hypotrochoid is a hypocycloid and p is a cusp of it, and (c) f has more than two roots with norm v if and only if p = 0 if and only if the hypotrochoid is a rhodonea curve with a multiple point of multiplicity s + t at the origin.
منابع مشابه
Several Classes of Permutation Trinomials over $\mathbb F_{5^n}$ From Niho Exponents
The construction of permutation trinomials over finite fields attracts people’s interest recently due to their simple form and some additional properties. Motivated by some results on the construction of permutation trinomials with Niho exponents, by constructing some new fractional polynomials that permute the set of the (q + 1)-th roots of unity in Fq2 , we present several classes of permutat...
متن کاملDescartes’ Rule for Trinomials in the Plane and Beyond
We prove that any pair of bivariate trinomials has at most 5 isolated roots in the positive quadrant. The best previous upper bounds independent of the polynomial degrees counted only non-degenerate roots and even then gave much larger bounds, e.g., 248832 via a famous general result of Khovanski. Our bound is sharp, allows real exponents, and extends to certain systems of n-variate fewnomials....
متن کاملm at h . C O ] 9 A ug 2 00 0 Descartes ’ Rule for Trinomials in the Plane and Beyond ∗
We prove that any pair of bivariate trinomials has at most 16 roots in the positive quadrant, assuming there are only finitely many roots there. The best previous upper bound independent of the polynomial degrees (following from a general result of Khovanski with stronger non-degeneracy hypotheses) was 248,832. Our proof allows real exponents and extends to certain systems of n-variate fewnomials.
متن کاملRoot Separation for Trinomials
We give a separation bound for the complex roots of a trinomial f ∈ Z[X ]. The logarithm of the inverse of our separation bound is polynomial in the size of the sparse encoding of f ; in particular, it is polynomial in log(deg f). It is known that no such bound is possible for 4-nomials (polynomials with 4 monomials). For trinomials, the classical results (which are based on the degree of f rat...
متن کاملDescartes’ Rule for Trinomials in the Plane and Beyond
We prove that any pair of bivariate trinomials has at most 5 isolated roots in the positive quadrant. The best previous upper bounds independent of the polynomial degrees counted only non-degenerate roots and even then gave much larger bounds, e.g., 248832 via a famous general result of Khovanski. Our bound is sharp, allows real exponents, and extends to certain systems of n-variate fewnomials,...
متن کامل