aa r X i v : . [ m a t h . G M ] S e p New Bounds on the Real PolynomialRoots
Emil M. Prodanov
School of Mathematical Sciences, Technological University Dublin,City Campus, Kevin Street, Dublin, D08 NF82, Ireland,E-Mail: [email protected]
Abstract
The presented analysis determines several new bounds on the roots of the equation a n x n + a n − x n − + · · · + a = 0 (with a n > { , P n − j =0 | a j /a n |} . Firstly, the Cauchy boundformula is derived by presenting it in a new light — through a recursion. It is shownthat this recursion could be exited at earlier stages and, the earlier the recursion isterminated, the lower the resulting root bound will be. Following a separate analysis,it is further demonstrated that a significantly lower root bound can be found if thesummation in the Cauchy bound formula is made not over each one of the coefficients a , a , . . . , a n − , but only over the negative ones. The sharpest root bound in thisline of analysis is shown to be the larger of 1 and the sum of the absolute valuesof all negative coefficients of the equation divided by the largest positive coefficient.The following bounds are also found in this paper: max { , ( P qj =1 B j /A l ) / ( l − k ) } ,where B , B , . . . B q are the absolute values of all of the negative coefficients in theequation, k is the highest degree of a monomial with a negative coefficient, A l is thepositive coefficient of the term A l x l for which k < l ≤ n . Mathematics Subject Classification Codes (2020) : 12D10, 26C10.
Keywords : Polynomial equation, Root bounds, Cauchy polynomial, Cauchy theorem,Cauchy and Lagrange bounds. 1he roots of the equation a n x n + a n − x n − + · · · + a = 0 are bound from above by theunique positive root of the associated Cauchy polynomial | a n | x n − | a n − | x n − − · · · −| a | . The Cauchy formula yields that the upper bound of the roots of the equation ismax { , P n − j =0 | a j /a n |} — see section 8.1 in [1]. In the first part of the analysis presentedin this paper, this formula is derived from a different perspective — through a recursion,— following the idea of splitting polynomials into two parts and studying the “interac-tion” between these parts [2] (i.e. studying the intersection points of their graphs) — amethod successfully used also for the full classification of the roots of the cubic [2] andthe quartic equation [3] in terms of the equation coefficients. This recursion involvesbounding the unique positive root of a particular equation with the unique positive rootof a subsidiary equation of degree one less. The recursion ends with the determination ofthe root of a linear equation and this root is exactly P n − j =0 | a j /a n | . If, instead, the recur-sion is terminated at an earlier stage — that of a quadratic, cubic, or quartic equation— the resulting root bound will be lower, as shown in this work. All of these analyticallydeterminable new bounds are lower than the Cauchy bound.It is separately shown, following a different line of analysis, that one does not have to sumover all coefficients a j in P n − j =0 | a j /a n | , but only over the negative ones. This results ina significantly lower root bound that the Cauchy bound. It is demonstrated further thatthis new bound can be made even lower by finding a denominator, greater than | a n | . Thesharpest upper bound of the roots of the general polynomial a n x n + a n − x n − + · · · + a with a n > { , ( P qj =1 B j /A l ) / ( l − k ) } , where B , B , . . . B q are the absolute values of all of the negative coefficients in the equation, k is the highest degree of a monomial with a negative coefficient, A l is the positive co-efficient of the term A l x l with k < l ≤ n .The lower bound on the unique positive root of a Cauchy polynomial is also determined.A Cauchy polynomial of degree n is defined [1] as a polynomial in x such that thecoefficient of x n is positive and the coefficients of all of its remaining terms — negative.That is, the Cauchy polynomial has the form | a n | x n − | a n − | x n − − · · · − | a | . As theremust be at least one term with negative coefficient, a monomial cannot be a Cauchypolynomial.The Cauchy polynomial has, by Descartes’ rule of signs, a unique positive root, say µ (as there is only one sign change in the sequence of its coefficients), and, by Cauchy’stheorem [1], the root µ provides the Cauchy bound on the roots of the general polynomialequation of degree n a n x n + a n − x n − + · · · + a = 0 . (1)2n this equation a n = 0 and, without losing generality, it will be assumed further that a n > α n x n − α n − x n − − · · · − α , where all coefficients α j are positive (the number of terms α j x j forwhich j < n may be between 1 and n ), and consider the associated equation α n x n − α n − x n − − · · · − α = 0 . (2)As the unique positive root of this equation is sought, one can assume, without loss ofgenerality, that α = 0. (If α happens to be zero, one identifies 0 as a root, reduceseverywhere the powers of x and the indexes of the coefficients by one unit and arrivesat an equation of the same type, but of degree n −
1. If the “new” α also happens tobe zero, this procedure should be repeated until a non-zero coefficient is encountered —it is guaranteed to exist by the definition.)Equation (2) can be viewed as f n ( x ) = α , (3)where f n ( x ) ≡ α n x n − α n − x n − − · · · − α x , and also as xf n − ( x ) = α , (4)where f n − ( x ) ≡ α n x n − − α n − x n − − · · · − α (since x = 0).Due to f n ( x ) = xf n − ( x ), the two polynomials f n ( x ) and f n − ( x ) have the same uniquepositive root r , that is, the graphs of the two functions f n ( x ) and f n − ( x ) cross eachother at point r on the abscissa.There is another intersection point between f n ( x ) and f n − ( x ) for positive x and thisintersection always happens at x = 1. This can be easily seen from f n (1) = α n − α n − − · · · − α = f n − (1) . (5)Thus, the coordinates of the two points of intersection between f n ( x ) and f n − ( x ) are( r,
0) and (1 , y ), where y = α n − α n − − · · · − α . Next, one has to determine wherepoint y is with respect to the “level” y = α >
0, prescribed by the free term of theCauchy polynomial.Clearly, when y = α n − α n − − · · · − α <
0, this intersection point is in the fourthquadrant, when y = α n − α n − − · · · − α >
0, it is in the first quadrant, and when y = α n − α n − − · · · − α = 0, then the functions f n ( x ) and f n − ( x ) cross only once atthe abscissa at r = 1, i.e. r = 1 is a double root of f n ( x ) = f n − ( x ) .As α >
0, there are three possible situations: either 0 ≤ y ≤ α (Figure 1), or y ≤ ≤ α (Figure 2), or 0 < α < y (Figure 3).3 igure 1 Figure 2 Figure 30 ≤ y ≤ α When α ≥ α n − α n − − · · · − α ≥
0, the upper bound of µ n is µ n − and the lower bound of µ n is 1. y ≤ ≤ α When α ≥ ≥ α n − α n − −· · · − α , the upper bound of µ n is µ n − and the lower bound of µ n is 1. 0 < α < y When 0 < α < α n − α n − −· · · − α , the upper bound of µ n is 1 and the lower bound of µ n is µ n − . Let µ n denote the unique positive root of f n ( x ) = α (note that µ = µ n ) and µ n − — theunique positive root of f n − ( x ) = α . It is quite clear that in the first two cases (Figures1 and 2) one has 1 < µ n < µ n − , while in the third case (Figure 3), µ n − < µ n < α ≥ α n − α n − − · · · − α , one has that µ n − is an upper bound of µ n (or µ )and 1 is a lower bound µ n (or µ ). On the contrary, when α < α n − α n − − · · · − α , onehas that µ n − is the lower bound of µ n (or µ ) and 1 is the upper bound µ n (or µ ). Inthe last case, one does not need to proceed further and should just take 1 as the upperbound of the unique positive root µ of the Cauchy polynomial and, hence, as the upperbound of the roots of the general equation (1).If, however, the free term α of the Cauchy polynomial is such that a ≥ α n − α n − −· · · − α , one needs to find an upper bound of µ n − and this, in turn, will serve as upperbound of µ n = µ and, hence, on the roots of (1). This means to continue recursively byconsidering the equation f n − ( x ) = α and re-writing it as g n − ( x ) = α ′ (6)where g n − ( x ) = α n x n − − α n − x n − − · · · − α x = f n − ( x ) + α and α ′ = α + α , onone hand, and as xg n − ( x ) = α ′ , (7)with g n − ( x ) = α n x n − − α n − x n − − · · · − α , on the other hand.As before, since g n − ( x ) = xg n − ( x ), the polynomials g n − ( x ) and g n − ( x ) have the sameunique positive root r ′ , that is, the graphs of the two functions g n − ( x ) and g n − ( x ) crosseach other at point r ′ on the abscissa. Also as before, there is another intersection pointbetween g n − ( x ) and g n − ( x ) for positive x and this intersection point is again x = 1: g n − (1) = α n − α n − − · · · − α = g n − (1) . (8)4quations (6) and (7) have the same unique positive root µ n − . Let µ n − denote theunique positive root of g n − ( x ) = α ′ . This positive root is an upper bound for µ n − provided that α ′ ≥ α n − α n − − · · · − α . The latter is simply a ≥ α n − α n − − · · · − α and this is indeed the case, as it was presumed to hold when the recursion started.The equation g n − ( x ) = α ′ is, in fact, α n x n − − α n − x n − − · · · − α x − α − α = α . (9)Continuing in the vein of recursively bounding the root of each of these equations withthe root of an equation of degree reduced by one unit, the linear equation α n x − α n − − α n − − · · · − α = 0 , (10)which terminates the recursion, immediately follows. The exact unique positive root ofthis equation is µ = α n − + α n − + · · · + α α n . (11)Thus, one naturally arrives at the Cauchy bound ρ = max { , µ } = max , n − X j =0 α j α n (12)for the roots of the general equation (1) — see (8.1.10) in [1] (there are no absolutevalues in the above formula as the α ’s are taken as positive).Next, sharper bounds than (12) will be found.The recursion which led to (11) could be exited earlier. For example, if one ends at thestage of quadratic equation (preceding that of the linear equation whose root is µ ), theunique positive root µ of this quadratic equation will be smaller than µ and hence,it will provide a sharper bound. That is, a sharper bound is provided by the uniquepositive root of α n x − α n − x − α n − − · · · − α = 0 , (13)namely µ = 12 α n (cid:20) α n − + q α n − + 4 α n ( α n − + α n − + · · · + α ) (cid:21) . (14)The recursion can be exited earlier than this — at the stage of the cubic equation α n x − α n − x − α n − x − α n − − · · · − α = 0 . (15)Its unique positive root µ which is such that µ < µ < µ , provides an even sharperbound.Finally, µ , the unique positive root of the quartic equation α n x − α n − x − α n − x − α n − x − α n − − · · · − α = 0 (16)5rovides the sharpest bound that can be found analytically by the recursion.The Cauchy bound (12) can be made significantly sharper by following a different lineof analysis.Suppose that the number of terms with positive coefficients in the general equation (1)is p and that the number of terms with negative coefficients is q . Clearly, p + q ≤ n + 1(equality is achieved if none of the coefficients of the general equation (1) is equal tozero). For the coefficients a j >
0, write A j instead, and for the coefficients a j <
0, write( − B j ) instead. Clearly, A n ≡ a n > A ’s are non-negative. Atleast one of the B ’s is positive, the rest — non-negative (equation in which all coefficientshave the same sign is no longer of interest).Following the ideas, presented in [2], of splitting a polynomial into two parts andanalysing the “interaction” between these parts in order to study the roots of the poly-nomial, one can re-write the general equation (1) as A l x l − B n − m x n − m − B n − m x n − m − · · · − B n − m q x n − m q = − A n x n − A n − k x n − k − · · · − ˆ A l x l − · · · − A n − k p − x n − k p − , (17)where { k < k < . . . < k p − , m < m < . . . < m q } is a permutation of { , , . . . , n } , l is such that n − m < l ≤ n , and the hat on A l indicates that the term A l x l is missingfrom the right-hand side. At least one monomial A l x l with n − m < l ≤ n exists — A n x n .The polynomial on left-hand side of (17) is a Cauchy polynomial with unique positiveroot µ (if the free term of the equation happens to be positive, then the resulting Cauchypolynomial will have zero as a root). This polynomial, due to having a positive coefficientin its leading term, is strictly positive for all x > µ . The polynomial on the right-handside of the equation is strictly negative for all x > x > µ , the two sides have opposite sign and therefore, there can be noroots of the equation for x > µ , i.e. the root bound for the general equation (1) is µ —the unique positive root of the Cauchy polynomial extracted from the given polynomial.Of course, different choices of l in A l x l will lead to different Cauchy polynomials withdifferent roots on the left-hand side of (17). It will be the unique positive root of theparticular Cauchy polynomial appearing on the left-hand side of (17) that will providea bound for the roots of the general equation (1).Suppose now that on the left-hand side of (17) the term − B l − x l − − B l − x l − − · · · − B m − n +1 x n − m +1 with B l − = B l − = . . . = B m − n +1 = 0 has been added to the Cauchypolynomial. Following the analysis that lead to the derivation of (11) and (12), it can beseen that, instead of the bound given by the Cauchy formula (12), a significantly lowerbound can be used ρ ′ l = max , q X j =1 B n − m j A l . (18)The summation is no longer over all coefficients of the equation (as in the Cauchyformula), but only over the absolute values of the negative ones (in units of A l ). There6ill be different bounds ρ ′ l for different choices of l .Clearly, the sharpest of these bounds will be the one with the biggest denominator: A max = max { A l | n ≥ l > n − m } : ρ ′ = max , q X j =1 B n − m j A max . (19)It should also be noted that, instead of introducing the “ghost” term B l − x l − − B l − x l − −· · ·− B m − n +1 with B l − = B l − = . . . = B m − n +1 = 0, a sharper bound can be obtainedif one terminates the recursion when the equation A l x l − n − m − B n − m − B n − m − · · · − B n − m q = 0 (20)is reached. The obtained in such way bound, for different values of l , is ρ ′′ l = max , q X j =1 B n − m j A l l − n − m , n − m < l ≤ n. (21)From here, one can find ρ ′′ as the minimum of the above.The above arguments prove the following theorem An upper bound of the roots of the general polynomial a n x n + a n − x n − + · · · + a with a n > is the smallest of the unique positive roots of all Cauchypolynomials that can be extracted form this polynomial with preservation of allterms with negative coefficients. To determine the lower bound of the unique positive root of the Cauchy polynomial,re-write equation (2) for positive x as x n (cid:18) α n − α n − x − · · · − α x n (cid:19) = 0 . (22)In variable y = 1 /x , this equation becomes: α y n + α y n − + · · · + α n − y − α n = 0 . (23)As there are n Cauchy polynomials α n − t y t − α n ( t = 1 , , . . . , n ) that can be extractedfrom α y n + α y n − + · · · + α n − y − α n , equation (23) can be written in n equivalentways: − α n + α n − t y t = − α y n − α y n − − · · · − ˆ α n − t y t − · · · − α n − y. (24)For each of these, the unique positive root ( α n /α n − t ) /t of the Cauchy polynomial α n − t y t − α n provides an upper bound on the roots of equation (23). The full set ofthe obtained in this way bounds on the roots of this equation is ( α n α n − , (cid:18) α n α n − (cid:19) , . . . , (cid:18) α n α (cid:19) n ) . (25)7hus, the sharpest upper bound on the roots of equation (23) is, obviously, the smallestof all these numbers. This is a Lagrange type of bound (the Lagrange bound is the sumof the two largest values of this sequence).The sharpest lower bound of the unique positive root of the Cauchy equation (2) willthus be the largest of the reciprocals of the numbers in the sequence (25). References [1] Q.I. Rahman and G. Schmeisser,
Analytic Theory of Polynomials , Oxford UniversityPress (2002).[2] Emil M. Prodanov,
On the Determination of the Number of Positive and NegativePolynomial Zeros and Their Isolation , arXiv: 1901.05960.[3] Emil M. Prodanov,