In mathematics, Vincent's theorem—named after —is a theorem that isolates the real roots of polynomials with rational coefficients.
Even though Vincent's theorem is the basis of the fastest method for the isolation of the real roots of polynomials, it was almost totally forgotten, having been overshadowed by Sturm's theorem; consequently, it does not appear in any of the classical books on the theory of equations (of the 20th century), except for Uspensky's book. Two variants of this theorem are presented, along with several (continued fractions and bisection) real root isolation methods derived from them.
Sign variation Let c0, c1, c2, ... be a finite or infinite sequence of real numbers. Suppose l < r and the following conditions hold: If r = l+1 the numbers cl and cr have opposite signs. If r ≥ l+2 the numbers cl+1, ..., cr−1 are all zero and the numbers cl and cr have opposite signs. This is called a sign variation or sign change between the numbers cl and cr. When dealing with the polynomial p(x) in one variable, one defines the number of sign variations of p(x) as the number of sign variations in the sequence of its coefficients.
Two versions of this theorem are presented: the continued fractions version due to Vincent, and the bisection version due to Alesina and Galuzzi.
Vincent's theorem: Continued fractions version (1834 and 1836) If in a polynomial equation with rational coefficients and without multiple roots, one makes successive transformations of the form
where are any positive numbers greater than or equal to one, then after a number of such transformations, the resulting transformed equation either has zero sign variations or it has a single sign variation. In the first case there is no root, whereas in the second case there is a single positive real root. Furthermore, the corresponding root of the proposed equation is approximated by the finite continued fraction:
Moreover, if infinitely many numbers satisfying this property can be found, then the root is represented by the (infinite) corresponding continued fraction.
The above statement is an exact translation of the theorem found in Vincent's original papers; however, the following remarks are needed for a clearer understanding: If denotes the polynomial obtained after n substitutions (and after removing the denominator), then there exists N such that for all either has no sign variation or it has one sign variation. In the latter case has a single positive real root for all . The continued fraction represents a positive root of the original equation, and the original equation may have more than one positive root. Moreover, assuming , we can only obtain a root of the original equation that is > 1. To obtain an arbitrary positive root we need to assume that . Negative roots are obtained by replacing x by −x, in which case the negative roots become positive.
Vincent's theorem: Bisection version (Alesina and Galuzzi 2000) Let p(x) be a real polynomial of degree deg(p) that has only simple roots. It is possible to determine a positive quantity δ so that for every pair of positive real numbers a, b with , every transformed polynomial of the form
has exactly 0 or 1 sign variations. The second case is possible if and only if p(x) has a single root within (a, b).
The Alesina–Galuzzi "a_b roots test"
From equation () the following criterion is obtained for determining whether a polynomial has any roots in the interval (a, b):
Perform on p(x) the substitution
and count the number of sign variations in the sequence of coefficients of the transformed polynomial; this number gives an upper bound on the number of real roots p(x) has inside the open interval (a, b). More precisely, the number ρab(p) of real roots in the open interval (a, b)—multiplicities counted—of the polynomial p(x) in R[x], of degree deg(p), is bounded above by the number of sign variations varab(p), where
As in the case of Descartes' rule of signs if varab(p) = 0 it follows that ρab(p) = 0 and if varab(p) = 1 it follows that ρab(p) = 1.
A special case of the Alesina–Galuzzi "a_b roots test" is Budan's "0_1 roots test".
Sketch of a proof A detailed discussion of Vincent's theorem, its extension, the geometrical interpretation of the transformations involved and three different proofs can be found in the work by Alesina and Galuzzi. A fourth proof is due to Ostrowski who rediscovered a special case of a theorem stated by Obreschkoff, p. 81, in 1920–1923.
To prove (both versions of) Vincent's theorem Alesina and Galuzzi show that after a series of transformations mentioned in the theorem, a polynomial with one positive root eventually has one sign variation. To show this, they use the following corollary to the theorem by Obreschkoff of 1920–1923 mentioned earlier; that is, the following corollary gives the necessary conditions under which a polynomial with one positive root has exactly one sign variation in the sequence of its coefficients; see also the corresponding figure.
Corollary. (Obreschkoff's cone or sector theorem, 1920–1923 p. 81): If a real polynomial has one simple root , and all other (possibly multiple) roots lie in the sector
then the sequence of its coefficients has exactly one sign variation.
Consider now the Möbius transformation
and the three circles shown in the corresponding figure; assume that
The (yellow) circle
whose diameter lies on the real axis, with endpoints and is mapped by the inverse Möbius transformation
onto the imaginary axis. For example the point
gets mapped onto the point The exterior points get mapped onto the half-plane with .
The two circles (only their blue crescents are visible) with center
and radius
are mapped by the inverse Möbius transformation
onto the lines . For example the point
gets mapped to the point
The exterior points (those outside the eight-shaped figure) get mapped onto the sector.
From the above it becomes obvious that if a polynomial has a single positive root inside the eight-shaped figure and all other roots are outside of it, it presents one sign variation in the sequence of its coefficients. This also guarantees the termination of the process.
Historical background
Early applications of Vincent's theorem In his fundamental papers, Vincent presented examples that show precisely how to use his theorem to isolate real roots of polynomials with continued fractions. However the resulting method had exponential computing time, a fact that mathematicians must have realized then, as was realized by Uspensky p. 136, a century later.
The exponential nature of Vincent's algorithm is due to the way the partial quotients ai (in Vincent's theorem) are computed. That is, to compute each partial quotient ai (that is, to locate where the roots lie on the x-axis) Vincent uses Budan's theorem as a "no roots test"; in other words, to find the integer part of a root Vincent performs successive substitutions of the form x ← x+1 and stops only when the polynomials p(x) and p(x+1) differ in the number of sign variations in the sequence of their coefficients (i.e. when the number of sign variations of p(x+1) is decreased).
See the corresponding diagram where the root lies in the interval (5, 6). It can be easily inferred that, if the root is far away from the origin, it takes a lot of time to find its integer part this way, hence the exponential nature of Vincent's method. Below there is an explanation of how this drawback is overcome.
Disappearance of Vincent's theorem Vincent was the last author in the 19th century to use his theorem for the isolation of the real roots of a polynomial.
The reason for that was the appearance of Sturm's theorem in 1827, which solved the real root isolation problem in polynomial time, by defining the precise number of real roots a polynomial has in a real open interval (a, b). The resulting (Sturm's) method for computing the real roots of polynomials has been the only one widely known and used ever since—up to about 1980, when it was replaced (in almost all computer algebra systems) by methods derived from Vincent's theorem, the fastest one being the Vincent–Akritas–Strzeboński (VAS) method.
Serret included in his Algebra, pp 363–368, Vincent's theorem along with its proof and directed all interested readers to Vincent's papers for examples on how it is used. Serret was the last author to mention Vincent's theorem in the 19th century.
Comeback of Vincent's theorem In the 20th century Vincent's theorem cannot be found in any of the theory of equations books; the only exceptions are the books by Uspensky and Obreschkoff, where in the second there is just the statement of the theorem.
It was in Uspensky's book that Akritas found Vincent's theorem and made it the topic of his Ph.D. Thesis "Vincent's Theorem in Algebraic Manipulation", North Carolina State University, USA, 1978. A major achievement at the time was getting hold of Vincent's original paper of 1836, something that had eluded Uspensky—resulting thus in a great misunderstanding. Vincent's original paper of 1836 was made available to Akritas through the commendable efforts (interlibrary loan) of a librarian in the Library of the University of Wisconsin–Madison, USA.
Real root isolation methods derived from Vincent's theorem Isolation of the real roots of a polynomial is the process of finding open disjoint intervals such that each contains exactly one real root and every real root is contained in some interval. According to the French school of mathematics of the 19th century, this is the first step in computing the real roots, the second being their approximation to any degree of accuracy; moreover, the focus is on the positive roots, because to isolate the negative roots of the polynomial p(x) replace x by −x (x ← −x) and repeat the process.
The continued fractions version of Vincent's theorem can be used to isolate the positive roots of a given polynomial p(x) of degree deg(p). To see this, represent by the Möbius transformation
the continued fraction that leads to a transformed polynomial with one sign variation in the sequence of its coefficients. Then, the single positive root of f(x) (in the interval (0, ∞)) corresponds to that positive root of p(x) that is in the open interval with endpoints and . These endpoints are not ordered and correspond to M(0) and M(∞) respectively.
Therefore, to isolate the positive roots of a polynomial, all that must be done is to compute—for each root—the variables a, b, c, d of the corresponding Möbius transformation
that leads to a transformed polynomial as in equation (), with one sign variation in the sequence of its coefficients.
Crucial Observation: The variables a, b, c, d of a Möbius transformation
(in Vincent's theorem) leading to a transformed polynomial—as in equation ()—with one sign variation in the sequence of its coefficients can be computed: either by continued fractions, leading to the Vincent–Akritas–Strzebonski (VAS) continued fractions method, or by bisection, leading to (among others) the Vincent–Collins–Akritas (VCA) bisection method.
The "bisection part" of this all important observation appeared as a special theorem in the papers by Alesina and Galuzzi.
All methods described below (see the article on Budan's theorem for their historical background) need to compute (once) an upper bound, ub, on the values of the positive roots of the polynomial under consideration. Exception is the VAS method where additionally lower bounds, lb, must be computed at almost every cycle of the main loop. To compute the lower bound lb of the polynomial p(x) compute the upper bound ub of the polynomial and set .