This paper researches two volatility transmission phenomena that take place within (‘heat wave’) and between (‘meteor shower’) spot and futures markets of four precious metals—gold, silver, platinum and palladium. We create conditional volatilities by considering three types of Markov switching GARCH models in combination with three different distribution functions. Conditional volatilities are subsequently embedded in Markov switching mean model. We find that ‘heat wave’ effect is more intense than ‘meteor shower’ effect, and this applies for both spot and futures markets of all precious metals. The results indicate that ‘heat wave’ effect is more intense in high than in low volatility periods, and also this effect is stronger in futures markets than in spot markets. ‘Meteor shower’ effect is stronger in low volatility regime than in high volatility regime, which is particularly true for the futures markets. Rolling regression results are in line with switching parameters. In addition, we find that ‘meteor shower’ effect, from futures to spot market, is stronger when short-term futures are analysed vis-à-vis long-term futures.

An arbitrary univariate polynomial of nth degree has n sequences. The sequences are systematized into classes. All the values of the first class sequence are obtained by Newton’s polynomial of nth degree. Furthermore, the values of all sequences for each class are calculated by Newton’s identities. In other words, the sequences are formed without calculation of polynomial roots. The New-nacci method is used for the calculation of the roots of an nth-degree univariate polynomial using radicals and limits of successive members of sequences. In such an approach as is presented in this paper, limit play a catalytic–theoretical role. Moreover, only four basic algebraic operations are sufficient to calculate real roots. Radicals are necessary for calculating conjugated complex roots. The partial limitations of the New-nacci method may appear from the decadal polynomial. In the case that an arbitrary univariate polynomial of nth degree (n ≥ 10) has five or more conjugated complex roots, the roots of the polynomial cannot be calculated due to Abel’s impossibility theorem. The second phase of the New-nacci method solves this problem as well. This paper is focused on solving the roots of the quintic equation. The method is verified by applying it to the quintic polynomial with all real roots and the Degen–Abel polynomial, dating from 1821.

In this paper we shall give another proof of a special case of Gao’s theorem for generating integrally indecomposable polygons in the sense of Minkowski. The approach of proving this theorem will enable us to give an effective algorithm for construction integrally indecomposable convex integral polygons with arbitrary many vertices. In such a way, classes of absolute irreducible bivariate polynomials corresponding to those indecomposable Newton polygons are generated.

Starting from n roots of a univariate polynomial of n -th degree, combining the first, the second, and up to the n -th class, new polynomials are generated. Each class of polynomials obtained in such a way is fully determined by its roots. The arithmetic means of corresponding coefficients are proportional to the values of the coefficients obtained by the derivation of the initial univariate polynomial. This gives the basis for the algebraic approach to the polynomial derivation.