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Let a polynomial with integer coefficients be nice if

  1. this polynomial has distinct integer roots;
  2. its derivative has also integer roots.

For instance $$p(x)=x(x-9)(x-24),\\ p'(x)=3(x-4)(x-18)$$ is the smallest known nice cubic polynomial. Smallest here means a polynomial with the smallest diameter. The diameter of the polynomial $p$ is the diameter of the set of the roots of $p$ in the complex plane. But how to verify with the help of MA that there are no smaller ones?

yarchik
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    Did you mean distinct integer roots? Also, nonnegative roots? – Carl Woll Sep 03 '21 at 19:19
  • An example with negative roots: (x + 1) (x - 8) (x - 23) that is smaller than 216. – Carl Woll Sep 03 '21 at 19:29
  • @CarlWoll Yes, distinct is important, I edited the question. Also the "smallest" needs to be slightly redefined. – yarchik Sep 03 '21 at 19:53
  • The redefinition in terms of the diameter of the set of roots makes the question much harder. If the roots are bounded by a certain magnitude away from zero, a brute-force approach could be employed. But I'm not sure how to get Mathematica to rule out the possibility that a polynomial with roots $N + a$, $N + b$, and $N +c$ is "nice" for relatively small $a$, $b$, $c$ and arbitrarily large $N$. – Michael Seifert Sep 03 '21 at 19:57
  • @MichaelSeifert Actually, I would be happy to see a solution even of a relaxed problem. For instance, to find another polynomial not related to my example by scaling or shift of variable. – yarchik Sep 03 '21 at 20:03
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    Wait, no, I withdraw my objection. If the polynomial I mentioned is "nice", then so is a polynomial with roots $a$, $b$, and $c$; just make the substitution $x \to x + N$. – Michael Seifert Sep 03 '21 at 20:10

2 Answers2

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We can perform a brute-force search on all such polynomials with roots up to a certain magnitude. First note that if $p(x)$ is a "nice" polynomial with roots $x_1 < x_2 < x_3$, then so is $q(x)$, the polynomial with roots $0$, $x_2 - x_1$, and $x_3-x_1$. This means that we only need to look for polynomials with one root at 0 and the other two roots positive integers. Moreover, we can restrict our case-checking to cases with $x_3 \geq 2 x_2$, since if we have a "nice" polynomial with roots $0, x_2, x_3$, then the polynomial with roots $0, x_3-x_2, x_3$ is also "nice". (Make the substitution $x \to x_3 - x$ to see this.) Finally, we can multiply any "nice" polynomial by an overall factor to get another "nice" polynomial; so we only need to check monic polynomials.

Define a monic polynomial with roots $0, x_2, x_3$, and a function to find the roots of its derivative:

p[x_, x2_, x3_] = x (x - x2) (x - x3)
derroots[x2_, x3_] = x /. Solve[D[p[x, x2, x3], x] == 0, x]

Generate all polynomials with $x_3 \leq x_\text{max}$, and select all of them for which their derivative vanishes at integer points:

xmax = 100;
results = Select[Flatten[
   Table[{{x2, x3}, derroots[x2, x3]}, 
         {x2, 1, xmax}, {x3, 2 x2, xmax}], 1], 
       And @@ Map[IntegerQ, Last[#], {1}] &]

(* {{{9, 24}, {4, 18}}, {{15, 63}, {7, 45}}, {{18, 48}, {8, 36}}, {{21, 45}, {9, 35}}, 
   {{27, 72}, {12, 54}}, {{36, 96}, {16, 72}}, {{42, 90}, {18, 70}}} *)

Interpreting this output, here are all the "nice" cubic polynomials with a root diameter less than or equal to 100, up to translations and inversions: \begin{align*} p(x) &= x(x-9)(x-24) & p'(x) &= 3 (x-4)(x-18) \\ p(x) &= x(x-21)(x-45) & p'(x) &= 3 (x-9)(x-35) \\ p(x) &= x(x-18)(x-48) & p'(x) &= 3 (x-8)(x-36)\\ p(x) &= x(x-15)(x-63) & p'(x) &= 3 (x-7)(x-45) \\ p(x) &= x(x-27)(x-72) & p'(x) &= 3 (x-12)(x-54) \\ p(x) &= x(x-42)(x-90) & p'(x) &= 3 (x-18)(x-70) \\ p(x) &= x(x-36)(x-96) & p'(x) &= 3 (x-16)(x-72) \\ \end{align*}

The case found by the OP is indeed the one with the smallest root diameter.

We can also define "quasi-nice" polynomials as those whose roots are integers and whose extrema are rational numbers. To find these, we replace IntegerQ in the above code by Element[#, Rationals] &; this yields four "quasi-nice" cubic polynomials with a root diameter smaller than 24: \begin{align*} p(x) &= x(x-3)(x-8) & p'(x) &= (3x-4)(x-6) \\ p(x) &= x(x-7)(x-15) & p'(x) &= (x-3)(3x-35) \\ p(x) &= x(x-6)(x-16) & p'(x) &= (3x-8)(x-12) \\ p(x) &= x(x-5)(x-21) & p'(x) &= (3x-7)(x-15) \end{align*} The code also returns an addition 24 "quasi-nice" polynomials with root diameter between 24 and 100.

Michael Seifert
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Using @Michael's formulation of the problem, we can use Reduce to show that the given nice polynomial is smallest. Consider:

p[x_] := x(x-b)(x-c);

We want to find positive integers b and c such that p'[x] = 3(x-d)(x-e) with d and e integers. So:

eqns = Resolve[ForAll[x, p'[x] == 3(x-d)(x-e)]]

2 b + 2 c - 3 d - 3 e == 0 && b c - 3 d e == 0

Now, using this we can check using Reduce:

Reduce[eqns && 0 < b < c < 24, {b, c, d, e}, Integers]

False

So, 24 is indeed the smallest diameter. We can find the OP polynomial with:

Reduce[eqns && 0 < b < c <= 24, {b, c, d, e}, Integers]

(b == 9 && c == 24 && d == 4 && e == 18) || (b == 9 && c == 24 && d == 18 && e == 4) || (b == 15 && c == 24 && d == 6 && e == 20) || (b == 15 && c == 24 && d == 20 && e == 6)

Carl Woll
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