# On the Hyperbolicity Domain of the Polynomial x^n + a1x^(n-1) + 1/4+ an

• Volume: 25, Issue: 1, page 47-70
• ISSN: 1310-6600

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## Abstract

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∗ Partially supported by INTAS grant 97-1644We consider the polynomial Pn = x^n + a1 x^(n−1) + · · · + an , ai ∈ R. We represent by figures the projections on Oa1 . . . ak , k ≤ 6, of its hyperbolicity domain Π = {a ∈ Rn | all roots of Pn are real}. The set Π and its projections Πk in the spaces Oa1 . . . ak , k ≤ n, have the structure of stratified manifolds, the strata being defined by the multiplicity vectors. It is known that for k > 2 every non-empty fibre of the projection Π^k → Π^(k−1) is a segment or a point. We prove that this is also true for the strata of Π of dimension ≥ k. This implies that for any two adjacent strata there always exist a space Oa1 . . . ak , k ≤ n, such that from the projections of the strata in it one is “above” the other w.r.t. the axis Oak . We show 1) how to find this k and which stratum is “above” just by looking at the multiplicity vectors of the strata; 2) how to obtain the relative position of a stratum of dimension l and of all strata of dimension l + 1 and l + 2 to which it is adjacent.

## How to cite

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Kostov, Vladimir. "On the Hyperbolicity Domain of the Polynomial x^n + a1x^(n-1) + 1/4+ an." Serdica Mathematical Journal 25.1 (1999): 47-70. <http://eudml.org/doc/11504>.

@article{Kostov1999,
abstract = {∗ Partially supported by INTAS grant 97-1644We consider the polynomial Pn = x^n + a1 x^(n−1) + · · · + an , ai ∈ R. We represent by figures the projections on Oa1 . . . ak , k ≤ 6, of its hyperbolicity domain Π = \{a ∈ Rn | all roots of Pn are real\}. The set Π and its projections Πk in the spaces Oa1 . . . ak , k ≤ n, have the structure of stratified manifolds, the strata being defined by the multiplicity vectors. It is known that for k > 2 every non-empty fibre of the projection Π^k → Π^(k−1) is a segment or a point. We prove that this is also true for the strata of Π of dimension ≥ k. This implies that for any two adjacent strata there always exist a space Oa1 . . . ak , k ≤ n, such that from the projections of the strata in it one is “above” the other w.r.t. the axis Oak . We show 1) how to find this k and which stratum is “above” just by looking at the multiplicity vectors of the strata; 2) how to obtain the relative position of a stratum of dimension l and of all strata of dimension l + 1 and l + 2 to which it is adjacent.},
journal = {Serdica Mathematical Journal},
keywords = {Hyperbolicity Domain; Stratum; Multiplicity Vector; hyperbolicity domain; stratum; multiplicity vector},
language = {eng},
number = {1},
pages = {47-70},
publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},
title = {On the Hyperbolicity Domain of the Polynomial x^n + a1x^(n-1) + 1/4+ an},
url = {http://eudml.org/doc/11504},
volume = {25},
year = {1999},
}

TY - JOUR
TI - On the Hyperbolicity Domain of the Polynomial x^n + a1x^(n-1) + 1/4+ an
JO - Serdica Mathematical Journal
PY - 1999
PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences
VL - 25
IS - 1
SP - 47
EP - 70
AB - ∗ Partially supported by INTAS grant 97-1644We consider the polynomial Pn = x^n + a1 x^(n−1) + · · · + an , ai ∈ R. We represent by figures the projections on Oa1 . . . ak , k ≤ 6, of its hyperbolicity domain Π = {a ∈ Rn | all roots of Pn are real}. The set Π and its projections Πk in the spaces Oa1 . . . ak , k ≤ n, have the structure of stratified manifolds, the strata being defined by the multiplicity vectors. It is known that for k > 2 every non-empty fibre of the projection Π^k → Π^(k−1) is a segment or a point. We prove that this is also true for the strata of Π of dimension ≥ k. This implies that for any two adjacent strata there always exist a space Oa1 . . . ak , k ≤ n, such that from the projections of the strata in it one is “above” the other w.r.t. the axis Oak . We show 1) how to find this k and which stratum is “above” just by looking at the multiplicity vectors of the strata; 2) how to obtain the relative position of a stratum of dimension l and of all strata of dimension l + 1 and l + 2 to which it is adjacent.
LA - eng
KW - Hyperbolicity Domain; Stratum; Multiplicity Vector; hyperbolicity domain; stratum; multiplicity vector
UR - http://eudml.org/doc/11504
ER -

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