Below are shown two displays of all the roots of polynomials $$c_n x^n + c_{n-1} x^{n-1} + \cdots + c_1 x + c_0 \;=\; 0$$ with each coefficient $c_i$ an integer $|c_i| \le c_{\max}$ (including $c_i=0$). No doubt this is all well-known, but I would be interested to learn what results explain the patterns in the distributions, especially the holes, both surrounding the real axis—in both shape and location—and the off-axis holes, perhaps more evident in the degree-$3$ plot than in the degree-$5$ plot.

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          [![roots_n3c6][1]][1]
          ^{Roots of polynomials of degree $\le 3$ and integer coefficients $|c_i| \le 6$.}

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          [![roots_n5c5][2]][2]
          ^{Roots of polynomials of degree $\le 5$ and integer coefficients $|c_i| \le 5$.}

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Below are shown two displays of all the roots of polynomials $$c_n x^n + c_{n-1} x^{n-1} + \cdots + c_1 x + c_0 \;=\; 0$$ with each coefficient $c_i$ an integer $|c_i| \le M$ (including $c_i=0$). No doubt this is all well-known, but I would be interested to learn what results explain the patterns in the distributions, especially the holes, both surrounding the real axis—in both shape and location—and the off-axis holes, perhaps more evident in the degree-$3$ plot than in the degree-$5$ plot.

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          [![roots_n3c6][1]][1]
          ^{Roots of polynomials of degree $\le 3$ and integer coefficients $|c_i| \le 6$.}

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          [![roots_n5c5][2]][2]
          ^{Roots of polynomials of degree $\le 5$ and integer coefficients $|c_i| \le 5$.}

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Addendum. User j.c. cited the article by John Baez on Dan Christensen's impressively detailed images, one of which I include below:

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[![enter image description here][3]][3]

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Below are shown two displays of all the roots of polynomials $$c_n x^n + c_{n-1} x^{n-1} + \cdots + c_1 x + c_0 \;=\; 0$$ with each coefficient $c_i$ an integer $|c_i| \le c_{\max}$. No doubt this is all well-known, but I would be interested to learn what results explain the patterns in the distributions, especially the holes, both surrounding the real axis—in both shape and location—and the off-axis holes, perhaps more evident in the degree-$3$ plot than in the degree-$5$ plot.

---

          [![roots_n3c6][1]][1]
          ^{Roots of polynomials of degree $\le 3$ and integer coefficients $|c_i| \le 6$.}

---

          [![roots_n5c5][2]][2]
          ^{Roots of polynomials of degree $\le 5$ and integer coefficients $|c_i| \le 5$.}

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Below are shown two displays of all the roots of polynomials $$c_n x^n + c_{n-1} x^{n-1} + \cdots + c_1 x + c_0 \;=\; 0$$ with each coefficient $c_i$ an integer $|c_i| \le c_{\max}$ (including $c_i=0$). No doubt this is all well-known, but I would be interested to learn what results explain the patterns in the distributions, especially the holes, both surrounding the real axis—in both shape and location—and the off-axis holes, perhaps more evident in the degree-$3$ plot than in the degree-$5$ plot.

---

          [![roots_n3c6][1]][1]
          ^{Roots of polynomials of degree $\le 3$ and integer coefficients $|c_i| \le 6$.}

---

          [![roots_n5c5][2]][2]
          ^{Roots of polynomials of degree $\le 5$ and integer coefficients $|c_i| \le 5$.}

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