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A Generalization of Habicht's Theorem for Subresultants of Several Univariate Polynomials

Hoon Hong, Jiaqi Meng, Jing Yang

Abstract

Subresultants of two univariate polynomials are one of the most classic and ubiquitous objects in computational algebra and algebraic geometry. In 1948, Habicht discovered and proved interesting relationships among subresultants. Those relationships were found to be useful for both structural understanding and efficient computation. Often one needs to consider several (possibly more than two) polynomials. It is rather straightforward to generalize the notion of subresultants to several polynomials. However, it is not obvious (in fact, quite challenging) to generalize the Habicht's result to several polynomials. The main contribution of this paper is to provide such a generalization.

A Generalization of Habicht's Theorem for Subresultants of Several Univariate Polynomials

Abstract

Subresultants of two univariate polynomials are one of the most classic and ubiquitous objects in computational algebra and algebraic geometry. In 1948, Habicht discovered and proved interesting relationships among subresultants. Those relationships were found to be useful for both structural understanding and efficient computation. Often one needs to consider several (possibly more than two) polynomials. It is rather straightforward to generalize the notion of subresultants to several polynomials. However, it is not obvious (in fact, quite challenging) to generalize the Habicht's result to several polynomials. The main contribution of this paper is to provide such a generalization.
Paper Structure (15 sections, 7 theorems, 114 equations)

This paper contains 15 sections, 7 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Review on subresultants of two polynomials
  4. Review on subresultants of several polynomials
  5. Main Result
  6. Relation to the Classical Habicht's Theorem
  7. Proof of the Generalized Habicht's Theorem (Theorem \ref{['thm:main']})
  8. Some useful properties of determinant polynomials
  9. Induction base (i.e. $k=1$)
  10. Proof of Eq. \ref{['k=1:conclusion1']}
  11. Proof of Eq. \ref{['k=1:conclusion2']}
  12. Inductive step (i.e. from $k=j$ to $k=j+1$)
  13. Proof of Eq. \ref{['equ:theorem:k=j+1_1']} from Eqs. \ref{['equ:theorem:k=j_1']}--\ref{['equ:theorem:k=j_2']}
  14. Proof of Eq. \ref{['equ:theorem:k=j+1_2']} from Eqs. \ref{['equ:theorem:k=j_1']}--\ref{['equ:theorem:k=j_2']}
  15. Conclusion

Key Result

Theorem 10

We have if $u\leq d_{0},v,w_{0},w_{1}, \epsilon$ satisfy the following conditions:

Theorems & Definitions (28)

  • Definition 1: Determinant polynomial of matrix
  • Example 2
  • Definition 3: Coefficient matrix of a list of polynomials
  • Example 4
  • Example 6
  • Definition 7
  • Example 8
  • Remark 9
  • Theorem 10: Habicht's Theorem Habicht:1948
  • Example 12
  • ...and 18 more