When does 6th degree polynomial have all 6 real roots?

Question

Does anyone know when the 6th degree polynomial has all 6 real roots? In which all coefficients are real number.

Thank you for your help!

Answer 1

Let $P(x)$ be the sextic polynomial at hand. For simplicity, I will assume you have simplified it to the form

$$P(x) = x^6 + ax^4 + bx^3 + cx^2 + dx + e$$

We can count the number of distinct real roots of $P(x)$ using Sturm's theorm.

Construct a sequence of polynomials $P_0,P_1,\ldots$ by the rule:

$$P_k(x) = \begin{cases} P,(x) & k = 0\\ P'(x), & k = 1\\ -{\rm rem}(P_{k-2}(x),P_{k-1}(x)), & k \ge 2 \end{cases} $$ where ${\rm rem}(P_{k-2}(x),P_{k-1}(x))$ is the remainder of the Euclidean division of $P_{k-2}(x)$ by $P_{k-1}(x)$.

Let $\alpha_k$ be the coefficient of $x^{6-k}$ in $P_k(x)$ for $0 \le k \le 6$. In order for $P(x)$ to have $6$ distinct real roots, the necessary and sufficient condition is all $\alpha_k$ are positive.

It is easy to see $\alpha_0 = 1$ and $\alpha_1 = 6$. This means in order for $P(x)$ to have $6$ distinct real roots, it need to satisfy $5$ polynomial inequalities. If I have copied the output from my CAS correctly, the $5$ inequalities can be chosen to be

$$R_5 = -12a > 0$$ $$ R_4 = 48ac-54b^2-16a^3 > 0 $$ $$\small R_3 = \left\{ \begin{array}{l} -288ace+324b^2e+96a^3e+300ad^2-720bcd-120a^2bd\\ +384c^3-224a^2c^2+324ab^2c+32a^4c-81b^4-12a^3b^2 \end{array}\right\} > 0 $$

$$ \small R_2 = \left\{ \begin{array}{l} -2592ae^3+6480bde^2+3456c^2e^2+864a^2ce^2-648ab^2e^2\\ -384a^4e^2-10800cd^2e+2160a^2d^2e+3024abcde-2268b^3de\\ -144a^3bde-2688ac^3e+1296b^2c^2e+1184a^3c^2e-1296a^2b^2ce\\ -128a^5ce+324ab^4e+48a^4b^2e+3750d^4-4800abd^3+2880ac^2d^2\\ +3240b^2cd^2-1232a^3cd^2+1116a^2b^2d^2+144a^5d^2-2688bc^3d\\ +864a^2bc^2d-972ab^3cd-112a^4bcd+162b^5d+24a^3b^3d+512c^5\\ -256a^2c^4+288ab^2c^3+32a^4c^3-54b^4c^2-8a^3b^2c^2 \end{array}\right\} > 0 $$

$$\small R_1 = \left\{ \begin{array}{l} -46656e^5+62208ace^4+34992b^2e^4-13824a^3e^4-32400ad^2e^3\\ -77760bcde^3+46656a^2bde^3-13824c^3e^3-17280a^2c^2e^3+3888ab^2ce^3\\ +9216a^4ce^3-8748b^4e^3-8640a^3b^2e^3-1024a^6e^3+27000bd^3e^2\\ +43200c^2d^2e^2-6480a^2cd^2e^2-27540ab^2d^2e^2-192a^4d^2e^2\\ -3456abc^2de^2+21384b^3cde^2-5760a^3bcde^2+5832a^2b^3de^2\\ +768a^5bde^2+9216ac^4e^2-8640b^2c^3e^2-4352a^3c^3e^2+8208a^2b^2c^2e^2\\ +512a^5c^2e^2-4860ab^4ce^2-576a^4b^2ce^2+729b^6e^2+108a^3b^4e^2\\ -22500cd^4e+1500a^2d^4e+19800abcd^3e-1350b^3d^3e-120a^3bd^3e\\ -10560ac^3d^2e-9720b^2c^2d^2e+4816a^3c^2d^2e-4536a^2b^2cd^2e\\ -576a^5cd^2e+162ab^4d^2e+24a^4b^2d^2e+6912bc^4de-2496a^2bc^3de\\ +2808ab^3c^2de+320a^4bc^2de-486b^5cde-72a^3b^3cde-1024c^6e\\ +512a^2c^5e-576ab^2c^4e-64a^4c^4e+108b^4c^3e+16a^3b^2c^3e\\ +3125d^6-3750abd^5+2000ac^2d^4+2250b^2cd^4-900a^3cd^4\\ +825a^2b^2d^4+108a^5d^4-1600bc^3d^3+560a^2bc^2d^3-630ab^3cd^3\\ -72a^4bcd^3+108b^5d^3+16a^3b^3d^3+256c^5d^2-128a^2c^4d^2\\ +144ab^2c^3d^2+16a^4c^3d^2-27b^4c^2d^2-4a^3b^2c^2d^2\end{array}\right\} > 0 $$

The last expression is so complicated that for practical application, it will be easier to write a routine to compute the Sturm's sequences directly.

If you just want to figure out whether a particular polynomial have $6$ distinct real roots, you can go the wolfram alpha and use the command subresultants there. For example, the command

subresultants[x^6+a*x^4+b*x^3+c*x^2+d*x+e,6*x^5+4*a*x^3+3*b*x^2+2*c*x+d,x]

will give you a list of $6$ entries $\{ -R_1, R_2, R_3, -R_4, -R_5 = 12a, 6 \}$ where $R_1,R_2,R_3,R_4,R_5$ is the LHS of above polynomial inequalities.

Once you get the list of $6$ entries from WA, the polynomial will have $6$ distinct real roots if the entries have the signs (-,+,+,-,-,+).