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In
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
, a branch of mathematics, the Gauss–Lucas theorem gives a
geometric Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
relation between the
root In vascular plants, the roots are the plant organ, organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often bel ...
s of a
polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
and the roots of its
derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
. The set of roots of a real or complex polynomial is a set of
points A point is a small dot or the sharp tip of something. Point or points may refer to: Mathematics * Point (geometry), an entity that has a location in space or on a plane, but has no extent; more generally, an element of some abstract topologica ...
in the
complex plane In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
. The theorem states that the roots of all lie within the
convex hull In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, ...
of the roots of , that is the smallest
convex polygon In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is ...
containing the roots of . When has a single root then this convex hull is a single point and when the roots lie on a line then the convex hull is a segment of this line. The Gauss–Lucas theorem, named after
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and ...
and Félix Lucas, is similar in spirit to
Rolle's theorem In calculus, Rolle's theorem or Rolle's lemma essentially states that any real-valued differentiable function that attains equal values at two distinct points must have at least one point, somewhere between them, at which the slope of the tangen ...
.


Formal statement

If is a (nonconstant) polynomial with complex coefficients, all zeros of belong to the convex hull of the set of zeros of .


Special cases

It is easy to see that if P(x) = ax^2+bx+c is a second degree polynomial, the zero of P'(x) = 2ax+b is the
average In colloquial, ordinary language, an average is a single number or value that best represents a set of data. The type of average taken as most typically representative of a list of numbers is the arithmetic mean the sum of the numbers divided by ...
of the roots of . In that case, the convex hull is the line segment with the two roots as endpoints and it is clear that the average of the roots is the middle point of the segment. For a third degree complex polynomial (
cubic function In mathematics, a cubic function is a function of the form f(x)=ax^3+bx^2+cx+d, that is, a polynomial function of degree three. In many texts, the ''coefficients'' , , , and are supposed to be real numbers, and the function is considered as ...
) with three distinct zeros,
Marden's theorem In mathematics, Marden's theorem, named after Morris Marden but proved about 100 years earlier by Jörg Siebeck, gives a geometric relationship between the zeroes of a third-degree polynomial with complex coefficients and the zeroes of its deriva ...
states that the zeros of are the foci of the Steiner inellipse which is the unique ellipse tangent to the midpoints of the triangle formed by the zeros of . For a fourth degree complex polynomial (
quartic function In algebra, a quartic function is a function (mathematics), function of the form :f(x)=ax^4+bx^3+cx^2+dx+e, where ''a'' is nonzero, which is defined by a polynomial of Degree of a polynomial, degree four, called a quartic polynomial. A ''qu ...
) with four distinct zeros forming a concave
quadrilateral In Euclidean geometry, geometry a quadrilateral is a four-sided polygon, having four Edge (geometry), edges (sides) and four Vertex (geometry), corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''l ...
, one of the zeros of lies within the convex hull of the other three; all three zeros of lie in two of the three triangles formed by the interior zero of and two others zeros of . In addition, if a polynomial of degree of real coefficients has distinct real zeros x_1 we see, using
Rolle's theorem In calculus, Rolle's theorem or Rolle's lemma essentially states that any real-valued differentiable function that attains equal values at two distinct points must have at least one point, somewhere between them, at which the slope of the tangen ...
, that the zeros of the derivative polynomial are in the interval _1,x_n/math> which is the convex hull of the set of roots. The convex hull of the roots of the polynomial : p_n x^n+p_x^+\cdots +p_0 particularly includes the point :-\frac.


Proof


See also

*
Marden's theorem In mathematics, Marden's theorem, named after Morris Marden but proved about 100 years earlier by Jörg Siebeck, gives a geometric relationship between the zeroes of a third-degree polynomial with complex coefficients and the zeroes of its deriva ...
*
Bôcher's theorem In mathematics, Bôcher's theorem is either of two theorems named after the American mathematician Maxime Bôcher. Bôcher's theorem in complex analysis In complex analysis, the theorem states that the finite zero of a function, zeros of the deriva ...
*
Sendov's conjecture In mathematics, Sendov's conjecture, sometimes also called Ilieff's conjecture, concerns the relationship between the locations of roots and critical points of a polynomial function of a complex variable. It is named after Blagovest Sendov. Th ...
*
Routh–Hurwitz theorem In mathematics, the Routh–Hurwitz theorem gives a test to determine whether all roots of a given polynomial lie in the left-half complex plane. Polynomials with this property are called Hurwitz stable polynomials. The Routh–Hurwitz theorem ...
*
Hurwitz's theorem (complex analysis) In mathematics and in particular the field of complex analysis, Hurwitz's theorem is a theorem associating the zeroes of a sequence of holomorphic, compact locally uniformly convergent functions with that of their corresponding limit. The theor ...
*
Descartes' rule of signs In mathematics, Descartes' rule of signs, described by René Descartes in his ''La Géométrie'', counts the roots of a polynomial by examining sign changes in its coefficients. The number of positive real roots is at most the number of sign chang ...
*
Rouché's theorem Rouché's theorem, named after Eugène Rouché, states that for any two complex-valued functions and holomorphic inside some region K with closed contour \partial K, if on \partial K, then and have the same number of zeros inside K, where ...
*
Properties of polynomial roots In mathematics, a univariate polynomial of degree with real or complex coefficients has complex ''roots'' (if counted with their multiplicities). They form a multiset of points in the complex plane, whose geometry can be deduced from the degre ...
* Cauchy interlacing theorem


Notes


References

* * . * * Craig Smorynski: ''MVT: A Most Valuable Theorem''. Springer, 2017, ISBN 978-3-319-52956-1, pp. 411–414


External links

*
Lucas–Gauss Theorem
by Bruce Torrence, the
Wolfram Demonstrations Project The Wolfram Demonstrations Project is an Open source, open-source collection of Interactive computing, interactive programmes called Demonstrations. It is hosted by Wolfram Research. At its launch, it contained 1300 demonstrations but has grown t ...
.
Gauss-Lucas theorem
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