02 / Maths Lab
Polynomial Zeros.
Sample an integer polynomial, then explore its complex zeros. Choose the convention from the report or the related paper.
MFO report · the radius convention
Sample an integer polynomial.
For n ∈ Z>0 and r > 0, the report defines
‖P‖B,∞ = max0≤k≤n |ak| / √C(n,k).
This set includes the zero polynomial, constants and polynomials of degree below n. Here C(n,k) = n!/[k!(n−k)!].
Choose n and r, then sample a polynomial.
Accumulate up to 100 polynomials under the same n and r. Changing either parameter starts a new collection.
The plot will show numerical approximations to the complex zeros.
Real and imaginary axes use the same scale. Roots are counted with multiplicity; coincident or nearby markers may overlap. The dashed unit circle is a reference.
Numerical zero coordinates
| Zero | Real part | Imaginary part |
|---|
How the sampling works
Each coefficient ak is chosen independently and uniformly from the integers −Bk, …, Bk, where Bk = ⌊r√C(n,k)⌋. Thus every member of 𝒫n,r has the same sampling probability. We keep zero and degree drops instead of drawing again.
Connection with the distribution theorem
The report proves convergence on average of normalized zero measures to the Fubini–Study measure when 1 < lim inf rn1/n ≤ lim sup rn1/n < ∞. A fixed raw radius does not satisfy this hypothesis. The related paper’s sampler uses a logarithmic parameter instead.
Shared experiments retain a repeatable pseudorandom sequence. Accumulation overlays finite numerical zero sets; it does not verify the asymptotic distribution theorem.
Corollary 1.8 · the logarithmic convention
Sample an integer polynomial.
For degree n ≥ 1, define the normalized Bombieri height
The paper uses the set
Here r > 0 is a logarithmic bound: the corresponding raw norm radius is enr. Unlike the report’s convention, the leading coefficient must be nonzero.
Choose n and r, then sample a polynomial.
Accumulate up to 100 polynomials under the same n and r. Changing either parameter starts a new collection.
The plot will show numerical approximations to the complex zeros.
Real and imaginary axes use the same scale. Roots are counted with multiplicity; coincident or nearby markers may overlap. The dashed unit circle is a reference.
Numerical zero coordinates
| Zero | Real part | Imaginary part |
|---|
How the sampling works
Put Bk = ⌊enr√C(n,k)⌋. Choose a0, …, an−1 independently and uniformly in −Bk, …, Bk; choose an uniformly from the 2Bn nonzero possibilities. This samples uniformly from the paper’s set. Irreducibility and primitivity are not imposed.
Connection with the distribution theorem
For fixed r > 0, Corollary 1.8 gives convergence on average of the normalized zero measures to the Fubini–Study measure as n grows. More generally, r may vary in a compact subinterval of (0,∞). The theorem concerns increasing degree; one finite sample need not resemble the limit.
Shared experiments retain a repeatable pseudorandom sequence. Accumulation overlays finite numerical zero sets; it does not verify the asymptotic distribution theorem.