26,486 research outputs found
Some remarks about Descartes' rule of signs
What can we deduce about the roots of a real polynomial in one variable by
simply considering the signs of its coefficients? On one hand, we give a
complete answer concerning the positive roots, by proposing a statement of
Descartes' rule of signs which strengthens the available ones while remaining
as elementary and concise as the original. On the other hand, we provide new
kinds of restrictions on the combined numbers of positive and negative roots.Comment: 10 page
Beyond Descartes' rule of signs
We consider real univariate polynomials with all roots real. Such a
polynomial with sign changes and sign preservations in the sequence of
its coefficients has positive and negative roots counted with
multiplicity. Suppose that all moduli of roots are distinct; we consider them
as ordered on the positive half-axis. We ask the question: If the positions of
the sign changes are known, what can the positions of the moduli of negative
roots be? We prove several new results which show how far from trivial the
answer to this question is
Descartes' rule of signs, Newton polygons, and polynomials over hyperfields
We develop a theory of multiplicities of roots for polynomials over
hyperfields and use this to provide a unified and conceptual proof of both
Descartes' rule of signs and Newton's "polygon rule".Comment: 21 pages. v2: Revised the exposition and organization of the paper,
corrected some minor typos. v3: Revised according to referee report, added
new reference
Descartes' Rule of Signs for Polynomial Systems supported on Circuits
We give a multivariate version of Descartes' rule of signs to bound the
number of positive real roots of a system of polynomial equations in n
variables with n+2 monomials, in terms of the sign variation of a sequence
associated both to the exponent vectors and the given coefficients. We show
that our bound is sharp and is related to the signature of the circuit.Comment: 25 pages, 3 figure
Polynomials that Sign Represent Parity and Descartes' Rule of Signs
A real polynomial sign represents if
for every , the sign of equals
. Such sign representations are well-studied in computer
science and have applications to computational complexity and computational
learning theory. In this work, we present a systematic study of tradeoffs
between degree and sparsity of sign representations through the lens of the
parity function. We attempt to prove bounds that hold for any choice of set
. We show that sign representing parity over with the
degree in each variable at most requires sparsity at least . We show
that a tradeoff exists between sparsity and degree, by exhibiting a sign
representation that has higher degree but lower sparsity. We show a lower bound
of on the sparsity of polynomials of any degree representing
parity over . We prove exact bounds on the sparsity of such
polynomials for any two element subset . The main tool used is Descartes'
Rule of Signs, a classical result in algebra, relating the sparsity of a
polynomial to its number of real roots. As an application, we use bounds on
sparsity to derive circuit lower bounds for depth-two AND-OR-NOT circuits with
a Threshold Gate at the top. We use this to give a simple proof that such
circuits need size to compute parity, which improves the previous bound
of due to Goldmann (1997). We show a tight lower bound of
for the inner product function over .Comment: To appear in Computational Complexit
New Aspects of Descartes’ Rule of Signs
Below, we summarize some new developments in the area of distribution of roots and signs of real univariate polynomials pioneered by R. Descartes in the middle of the seventeenth century
On Descartes' rule of signs for hyperbolic polynomials
We consider univariate real polynomials with all roots real and with two sign
changes in the sequence of their coefficients which are all non-vanishing. One
of the changes is between the linear and the constant term. By Descartes' rule
of signs, such degree polynomials have positive and negative
roots. We consider the sequences of the moduli of their roots on the real
positive half-axis. When the moduli are distinct, we give the exhaustive answer
to the question at which positions can the moduli of the two positive roots be
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