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

Serdica Mathematical Journal (1999)

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

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topKostov, 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.},

author = {Kostov, Vladimir},

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

AU - Kostov, Vladimir

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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