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David Savitt and some REU students write about "harmonic curves" and "meanders". I have not read the estimates in details and I heard they can be optimized.

• Harmonic algebraic curves and noncrossing partitions

• Polynomials, meanders, and paths in the lattice of noncrossing partitions

The idea is to check $\mathrm{Re}[p(x)]=0$ and $\mathrm{Im}[p(x)]=0$ and to show that they intersect at exactly $n = \mathrm{deg} \, p$ points. Then we can have a topological proof since we can split the plane into $\mathbb{C} \backslash \mathbb{D} \cup \mathbb{D}$ where on the outside we have an alternating series of lines extending out to infinity and inside we have some kind of non-crossing matching of the lines.

They come across an objection similar to what you have, is that topological picture correct? Can't the curves $I$ or $R$ be self-intersecting or worse?

Back then, there was no systematic study of shape such as topology. Savitt merely proves these curves are non-singular without explicitly ruling out these phenomena.

Then Savitt studies $\mathrm{Re}[e^{-i\theta}p(x)] = 0$ which contains $R$ and $I$ as special cases (set $\theta = 0, 90^\circ$)

David Savitt and some REU students write about "harmonic curves" and "meanders". I have not read the estimates in details and I heard they can be optimized.

• Harmonic algebraic curves and noncrossing partitions

• Polynomials, meanders, and paths in the lattice of noncrossing partitions

The idea is to check $\mathrm{Re}[p(x)]=0$ and $\mathrm{Im}[p(x)]=0$ and to show that they intersect at exactly $n = \mathrm{deg} \, p$ points. Then we can have a topological proof since we can split the plane into $\mathbb{C} \backslash \mathbb{D} \cup \mathbb{D}$ where on the outside we have an alternating series of lines extending out to infinity and inside we have some kind of non-crossing matching of the lines.

They come across an objection similar to what you have, is that topological picture correct? Can't the curves $I$ or $R$ be self-intersecting or worse?

Back then, there was no systematic study of shape such as topology. Savitt merely proves these curves are non-singular without explicitly ruling out these phenomena.

Then Savitt studies $\mathrm{Re}[e^{-i\theta}p(x)] = 0$ which contains $R$ and $I$ as special cases (set $\theta = 0, 90^\circ$)

David Savitt and some REU students write about "harmonic curves" and "meanders". I have not read the estimates in details and I heard they can be optimized.

• Harmonic algebraic curves and noncrossing partitions

• Polynomials, meanders, and paths in the lattice of noncrossing partitions

The idea is to check $\mathrm{Re}[p(x)]=0$ and $\mathrm{Im}[p(x)]=0$ and to shoe they intersect at exactly $n = \mathrm{deg} \, p$ points. Then we can have a topological proof since can split the plane into $\mathbb{C} \backslash \mathbb{D} \cup \mathbb{D}$ where on the outside we have an alternating series of lines extending out to infinity and inside we have some kind of non-crossing matching of the lines.

They come across an objection similar to what you have, is that topological picture correct? Can't the curves $I$ or $R$ be self-intersecting or worse?

Back then, there was no systematic study of shape such as topology. Savitt merely proves these curves are non-singular without explicitly ruling out these phenomena.

Then Savitt studies $\mathrm{Re}[e^{-i\theta}p(x)] = 0$ which contains $R$ and $I$ as special cases (set $\theta = 0, 90^\circ$)

David Savitt and some REU students write about "harmonic curves" and "meanders". I have not read the estimates in details and I heard they can be optimized.

• Harmonic algebraic curves and noncrossing partitions

• Polynomials, meanders, and paths in the lattice of noncrossing partitions

The idea is to check $\mathrm{Re}[p(x)]=0$ and $\mathrm{Im}[p(x)]=0$ and to show that they intersect at exactly $n = \mathrm{deg} \, p$ points. Then we can have a topological proof since we can split the plane into $\mathbb{C} \backslash \mathbb{D} \cup \mathbb{D}$ where on the outside we have an alternating series of lines extending out to infinity and inside we have some kind of non-crossing matching of the lines.

They come across an objection similar to what you have, is that topological picture correct? Can't the curves $I$ or $R$ be self-intersecting or worse?

Back then, there was no systematic study of shape such as topology. Savitt merely proves these curves are non-singular without explicitly ruling out these phenomena.

Then Savitt studies $\mathrm{Re}[e^{-i\theta}p(x)] = 0$ which contains $R$ and $I$ as special cases (set $\theta = 0, 90^\circ$)

David Savitt and some REU students write about "harmonic curves" and "meanders". I have not read the estimates in details and I heard they can be optimized.

• Harmonic algebraic curves and noncrossing partitions

• Polynomials, meanders, and paths in the lattice of noncrossing partitions

The idea is to check $\mathrm{Re}[p(x)]=0$ and $\mathrm{Im}[p(x)]=0$ and to shoe they intersect at exactly $n = \mathrm{deg} \, p$ points. Then we can have a topological proof since can split the plane into $\mathbb{C} \backslash \mathbb{D} \cup \mathbb{D}$ where on the outside we have an alternating series of lines extending out to infinity and inside we have some kind of non-crossing matching of the lines.

They come across an objection similar to what you have, is that topological picture correct? Can't the curves $I$ or $R$ be self-intersecting or worse?

Back then, there was no systematic study of shape such as topology. Savitt merely proves these curves are non-singular without explicitly ruling out these phenomena.

Then Savitt studies $\mathrm{Re}[e^{-i\theta}p(x)] = 0$ which contains $R$ and $I$ as special cases (set $\theta = 0, 90^\circ$)

David Savitt and some REU students write about "harmonic curves" and "meanders". I have not read the estimates in details and I heard they can be optimized.

• Harmonic algebraic curves and noncrossing partitions

• Polynomials, meanders, and paths in the lattice of noncrossing partitions

The idea is to check $\mathrm{Re}[p(x)]=0$ and $\mathrm{Im}[p(x)]=0$ and to shoe they intersect at exactly $n = \mathrm{deg} \, p$ points. Then we can have a topological proof since can split the plane into $\mathbb{C} \backslash \mathbb{D} \cup \mathbb{D}$ where on the outside we have an alternating series of lines extending out to infinity and inside we have some kind of non-crossing matching of the lines.

They come across an objection similar to what you have, is that topological picture correct? Can't the curves $I$ or $R$ be self-intersecting or worse?

Back then, there was no systematic study of shape such as topology. Savitt merely proves these curves are non-singular without explicitly ruling out these phenomena.

Then Savitt studies $\mathrm{Re}[e^{-i\theta}p(x)] = 0$ which contains $R$ and $I$ as special cases (set $\theta = 0, 90^\circ$)