Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Criteria for a split real polynomial

Jean-Michel Billiot, Eric Fontenas

Abstract

In this article, we establish necessary and sufficient conditions for a polynomial of degree $n$ to have exactly $n$ real roots. A complete study of polynomials of degree five is carried out. The results are compared with those obtained using Sturm sequences.

Criteria for a split real polynomial

Abstract

In this article, we establish necessary and sufficient conditions for a polynomial of degree to have exactly real roots. A complete study of polynomials of degree five is carried out. The results are compared with those obtained using Sturm sequences.
Paper Structure (15 sections, 12 theorems, 64 equations, 2 figures)

This paper contains 15 sections, 12 theorems, 64 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Method
  3. Discussion for polynomials of degree two and three
  4. General idea
  5. Main result
  6. Polynomial of order three and four
  7. Polynomial of order five
  8. Our assumptions
  9. Choice of $s$
  10. Study of the hypothesis of the theorem \ref{['teoP5']}
  11. Notes: degenerate cases
  12. Analysis of the first two cases
  13. The last case
  14. Comparison with Sturm's method
  15. Concluding remarks

Key Result

Proposition 1

1. Set $P_n(x)=xQ_{n-1}(x)-R_{n-2}(x)$. If $Q_{n-1}$ has $n-1$ real roots and if $R_{n-2}$ has $n-2$ real roots interlaced with those of $Q_{n-1}$, then $P_n$ has $n$ real roots. 2. For a polynomial of degree $n$, to have $n$ real roots, its derivative must have $n-1$ distinct real roots.

Figures (2)

  • Figure 1: an example of degenerate case with $a<b<c$
  • Figure 2: an example of the last degenerate case $a<c<b$

Theorems & Definitions (14)

  • Definition 1
  • Proposition 1
  • Proposition 2
  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Proposition 3
  • Theorem 3
  • Proposition 4
  • Proposition 5
  • ...and 4 more