WOLFRAM|DEMONSTRATIONS PROJECT

Descartes Signature Explorer

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sweep spirals
sweep graphs
Descartes signature
coefficient anglesdirections of polynomial coefficients
9
x
0.00
π
8
x
0.00
π
7
x
0.00
π
6
x
0.00
π
5
x
0.00
π
4
x
1.25
π
3
x
0.50
π
2
x
1.75
π
1
x
0.75
π
0
x
0.00
π
auxiliary angledirection of proposed root
θ
0
+
sweep
p
(θ) = 4.75 π
-
sweep
p
(θ) = 3.25 π
sweep
p
(θ) = 3.25 π
dsig
p
​(θ) = maximum number of roots in direction θ = 3.
The Descartes signature of a polynomial gives the maximum number of roots in each angular direction of the complex plane. It arises from the Descartes rule of sweeps, which extends the well-known rule of signs to polynomials with non-real coefficients. Whereas the rule of signs counts the sign changes of a polynomial's coefficient sequence, the rule of sweeps considers the polynomial's "angular sweep". This Demonstration provides a laboratory for investigating the relationships among coefficients, their sweeps, and the corresponding Descartes signature.