Remarks on algebraic equations whose derived equations have all roots real

Tiberiu Popoviciu
(original), Algebra, paper, polynomials

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T. Popoviciu

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T. Popoviciu, Remarques sur les équations algebriques dont les équations derivées ont toutes leurs racines réelles, Comptes Rendus de l’Acad. des Sci. de Paris, 200 (1935), pp. 184-186 (in French).

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1935 f -Popoviciu- Comptes Rendus Acad. Sci. Paris – Remarques sur les equations algebriques dont les equations derivees ont toutes leurs racines reelles

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Comptes Rendus de l’Acad. des Sci. de Paris

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Publisher: Académie des Sciences, Paris; Elsevier, Paris.

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[Zbl 0010.38801, JFM 61.0076.01]

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1935 f -Popoviciu- Comptes Rendus Acad. Sci. Paris - Remarks on algebraic equations of which the
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ALGEBRA. - Remarks on algebraic equations whose derivative equations all have real roots. Note by Mr. Tibere Popoviciu, presented by Mr. Hadamard.
  1. We only consider polynomials with real coefficients. Let P ( x ) P ( x ) P(x)\mathrm{P}(x)P(x)a polynomial of degree n n nnn. Let us designate by r r rrrthe number of real zeros that are multiples of the polynomial P ( x ) P ( x ) P(x)\mathrm{P}(x)P(x)and by k k kkkthe sum of the number of all distinct real zeros and the number of complex zeros of P ( x ) P ( x ) P(x)\mathrm{P}(\boldsymbol{x})P(x)Note that the number k + r − 1 k + r − 1 k+r-1k+r-1k+r−1is at most equal to n − 1 n − 1 n-1n-1n−1.
We obtained the following property:
If there exists a polynomial Q ( x ) Q ( x ) Q(x)Q(x)Q(x)of degree m m mmmsuch that the derivative equation [ P ( x ) ⋅ Q ( x ) ] ′ = o [ P ( x ) ⋅ Q ( x ) ] ′ = o [P(x)*Q(x)]^(')=o[\mathrm{P}(x) \cdot \mathrm{Q}(x)]^{\prime}=\mathrm{o}[P(x)⋅Q(x)]′=oGiven all its real roots, there certainly exists a polynomial R ( x ) R ( x ) R(x)\mathrm{R}(x)R(x)of degree ≦ m ≦ m <= m\leqq m≦msuch that the derivative equation [ P ( x ) ⋅ R ( x ) ] ′ = 0 [ P ( x ) ⋅ R ( x ) ] ′ = 0 [P(x)*R(x)]^(')=0[\mathrm{P}(x) \cdot \mathrm{R}(x)]^{\prime}=0[P(x)⋅R(x)]′=0has all its real roots, of which at most k + r − 1 k + r − 1 k+r-1k+r-1k+r−1are distinct.
The demonstration relies on the continuity of the roots of an algebraic equation with respect to the coefficients and, noting that, for m = 0 m = 0 m=0m=0m=0The property results from the fact that if a polynomial has a zero that is a multiple of order v ( v > 1 ) ( v > 1 ) (v > 1)(v>1)(v>1), its derivative has the same zero multiple of order v − 1 v − 1 v-1v-1v−1.
We can also say, without specifying the nature of the zeros of the polynomial P ( x ) P ( x ) P(x)\mathrm{P}(x)P(x)that if there exists a polynomial Q ( x ) Q ( x ) Q(x)Q(x)Q(x)such that the derivative equation [ P ( x ) ⋅ Q ( x ) ] ′ = 0 [ P ( x ) ⋅ Q ( x ) ] ′ = 0 [P(x)*Q(x)]^(')=0[\mathrm{P}(x) \cdot \mathrm{Q}(x)]^{\prime}=0[P(x)⋅Q(x)]′=0has all its real roots, there exists a polynomial R ( x ) R ( x ) R(x)\mathrm{R}(x)R(x)such that the derivative equation [ P ( x ) ⋅ R ( x ) ] ′ = 0 [ P ( x ) ⋅ R ( x ) ] ′ = 0 [P(x)*R(x)]^(')=0[\mathrm{P}(x) \cdot \mathrm{R}(x)]^{\prime}=0[P(x)⋅R(x)]′=0has all its real roots, of which at most n − 1 n − 1 n-1n-1n−1are distinct.
2. Either
F ( x ) = ( x − c ) p [ ( x − c ) 2 + d 2 ] Q ( x ) F ( x ) = ( x − c ) p ( x − c ) 2 + d 2 Q ( x ) F(x)=(xc)^(p)[(xc)^(2)+d^(2)]Q(x)\mathbf{F}(x)=(xc)^{p}\left[(xc)^{2}+d^{2}\right] \mathrm{Q}(x)F(x)=(x−c)p[(x−c)2+d2]Q(x)
Q ( x ) Q ( x ) Q(x)Q(x)Q(x)being a polynomial, c , d ≠ 0 c , d ≠ 0 c,d!=0c, d \neq 0c,d≠0real constants and p p pppa positive integer. Therefore:
The derivative equation Γ ′ ( x ) = 0 Γ ′ ( x ) = 0 Gamma^(')(x)=0\Gamma^{\prime}(x)=0Γ′(x)=0cannot have all its real roots.
It is again with the help of the continuity of roots that we have proven that it suffices to demonstrate the property for the case where F ′ ( x ) F ′ ( x ) F^(')(x)F^{\prime}(x)F′(x)would have at most two distinct zeros, one of which is always simple. In this case, the property is proven directly.
3. As a consequence of the previous property, we have the following theorem:
If the derived equation f ′ ( x ) = 0 f ′ ( x ) = 0 f^(')(x)=0f^{\prime}(x)=0f′(x)=0of an algebraic equation of degree n n nnnhas all its real roots and if the equation f ( x ) = 0 f ( x ) = 0 f(x)=0f(x)=0f(x)=0a pair of imaginary roots conjugated to ± i b ± i b +-ib\pm ib±ib, this equation cannot have any real roots in the interval
( has − b tang π n , has + b tang π n ) . has − b tang π n , has + b tang π n . (ab tang((pi )/(n)),a+b tang((pi )/(n))).\left(ab \operatorname{tang} \frac{\pi}{n}, a+b \operatorname{tang} \frac{\pi}{n}\right) .(has−btangπn,has+btangπn).
The limits are only reached for the equations
f ( x ) ≡ C [ ( x − has ± b cotg 2 π n ) n − ( ± b ) n ( sin ⁡ 2 π n ) n ] = 0 f ( x ) ≡ C x − has ± b cotg 2 π n n − ( ± b ) n sin ⁡ 2 π n n = 0 f(x)-=C[(x-a+-b cotg((2pi)/(n)))^(n)-((+-b)^(n))/((sin((2pi)/(n)))^(n))]=0f(x) \equiv \mathrm{C}\left[\left(xa \pm b \operatorname{cotg} \frac{2 \pi}{n}\right)^{n}-\frac{( \pm b)^{n}}{\left(\sin \frac{2 \pi}{n}\right)^{n}}\right]=0f(x)≡C[(x−has±bcotg2πn)n−(±b)n(sin⁡2πn)n]=0
Let C be a constant.
4. We also examined equations that have only one pair of imaginary conjugate roots and obtained the following theorem:
If the derivative equation f ′ ( x ) = 0 f ′ ( x ) = 0 f^(')(x)=0f^{\prime}(x)=0f′(x)=0of an algebraic equation of degree n n nnnhas all its real roots and if the equation f ( x ) = 0 f ( x ) = 0 f(x)=0f(x)=0f(x)=0a setil couple of imaginary roots conjugated a ± ± +-\pm±ib, this equation cannot have any roots in the interval
( has − λ n b , has + λ n b ) has − λ n b , has + λ n b (a-lambda_(n)b,a+lambda_(n)b)\left(a-\lambda_{n} b, a+\lambda_{n} b\right)(has−λnb,has+λnb)
Or λ n λ n lambda_(n)\lambda_{n}λnis the real and positive root of the equation
( n − 1 ) 2 ( n − 1 ) x ( x 2 + 9 ) − 3 ( n + 1 ) 3 ( n − 2 ) ( x 2 + 1 ) = 0 ( n − 1 ) 2 ( n − 1 ) x x 2 + 9 − 3 ( n + 1 ) 3 ( n − 2 ) x 2 + 1 = 0 (n-1)sqrt(2(n-1))x(x^(2)+9)-3(n+1)sqrt(3(n-2))(x^(2)+1)=0(n-1) \sqrt{2(n-1)} x\left(x^{2}+9\right)-3(n+1) \sqrt{3(n-2)}\left(x^{2}+1\right)=0(n−1)2(n−1)x(x2+9)−3(n+1)3(n−2)(x2+1)=0
The limits are reached for the equations
[ ( x − has ) 2 + b 2 ] ( x ± λ n ) ( x ± μ n ) n − 3 = 0 ( x − has ) 2 + b 2 x ± λ n x ± μ n n − 3 = 0 [(xa)^(2)+b^(2)](x+-lambda_(n))(x+-mu_(n))^(n-3)=0\left[(xa)^{2}+b^{2}\right]\left(x \pm \lambda_{n}\right)\left(x \pm \mu_{n}\right)^{n-3}=0[(x−has)2+b2](x±λn)(x±μn)n−3=0
where the signs correspond, and
μ n = − λ n [ ( n − 1 ) 2 λ n 2 + 3 ( 3 n 2 − 10 n − 1 ) ] 3 ( n + 1 ) ( 3 − λ n 2 ) . μ n = − λ n ( n − 1 ) 2 λ n 2 + 3 3 n 2 − 10 n − 1 3 ( n + 1 ) 3 − λ n 2 . mu_(n)=(-lambda_(n)[(n-1)^(2)lambda_(n)^(2)+3(3n^(2)-10 n-1)])/(3(n+1)(3-lambda_(n)^(2))).\mu_{n}=\frac{-\lambda_{n}\left[(n-1)^{2} \lambda_{n}^{2}+3\left(3 n^{2}-10 n-1\right)\right]}{3(n+1)\left(3-\lambda_{n}^{2}\right)} .μn=−λn[(n−1)2λn2+3(3n2−10n−1)]3(n+1)(3−λn2).
When n n nnnbelieves, λ n λ n lambda_(n)\lambda_{n}λndecreases and tends towards a positive limit λ λ lambda\lambdaλFor n → ∞ n → ∞ n rarr oon \rightarrow \inftyn→∞We can therefore state a similar property that does not depend on the degree n. The roots are outside an interval ( has − λ b , has + λ b has − λ b , has + λ b a-lambda b, a+lambda ba-\lambda b, a+\lambda bhas−λb,has+λb), Or λ λ lambda\lambdaλis the real and positive root of the equation
2 x ( x 2 + 9 ) − 3 3 ( x 2 + 1 ) = 0 2 x x 2 + 9 − 3 3 x 2 + 1 = 0 sqrt2x(x^(2)+9)-3sqrt3(x^(2)+1)=0\sqrt{2} x\left(x^{2}+9\right)-3 \sqrt{3}\left(x^{2}+1\right)=02x(x2+9)−33(x2+1)=0
The limits have not been reached, but the number λ λ lambda\lambdaλcannot be replaced by any other larger number.
λ λ lambda\lambdaλis close to 0.5. More precisely, it is between 0.4946 and 0.4947.
1935

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