Let p(t)∈Z≥0[t]. Write p(t)=∑iaiti. Then we say p(t) is unimodal if a0≤a1≤...≤ak≥ak+1≥...≥an i.e, the sequence of coefficients increase at first and then decrease, they don't 'jump around' ; there is no phenomenon like increase then decrease then increase again. Given such a polynomial, how can we check unimodality in sage?
Here is a way:
def is_unimodal(p):
coeffs = p.coefficients(sparse=False)
max_coeff_idx = max(range(len(coeffs)), key=lambda i: coeffs[i])
return all(coeffs[i] <= coeffs[i+1] for i in range(max_coeff_idx)) and \
all(coeffs[i] >= coeffs[i+1] for i in range(max_coeff_idx, len(coeffs) - 1))Example usage:
sage: R.<t> = ZZ[]
sage: p = R.random_element(degree=4).map_coefficients(abs); p
t^4 + 4*t^3 + t^2 + t + 1
sage: p.coefficients(sparse=False)
[1, 1, 1, 4, 1]
sage: is_unimodal(p)
TrueActually coeffs don't take into account if some coefficient ak=0 in the middle. So this works fine if there is no internal zeros, but if there is some then it gives wrong answer.
@mathstudent That wasn't written in your definition. I've updated the answer. Probably there is a faster way, but I guess speed doesn't matter much here.
No... I meant your earlier program didn't work if there was some internal zeroes
Ah sorry, I misunderstood you. Fixed now.
Thank you very much!
Call C the list of coefficients, removing immediate repeats.
Unimodality means there is no decrease-then-increase in C.
The package more-itertools (which can be installed with pip) enables an elegant solution:
def is_unimodal(p):
"""
Return whether this polynomial is unimodal.
EXAMPLES::
sage: R.<t> = ZZ[]
sage: is_unimodal(1 + t + t^2 + 4*t^3 + t^4) # coeffs: 1, 1, 1, 4, 1
True
sage: is_unimodal(1 + t^2 + 4*t^3 + t^4) # coeffs: 1, 0, 1, 4, 1
False
"""
from more_itertools import triplewise, unique_justseen
return all((a < b) or (b > c)
for a, b, c in triplewise(unique_justseen(p)))From @rburing's answer... slightly modified since taking coefficients in the polynomial ring somehow only gives the non-zero coefficients:
t = var('t')
R = ZZ[t]
def is_unimodal(p):
coeffs = [R(p).constant_coefficient()]
for i in range(1,p.degree(t)+1):
coeffs = coeffs + [SR(p).coefficient(t^i)]
max_coeff_idx = max(range(len(coeffs)), key=lambda i: coeffs[i])
return all(coeffs[i] <= coeffs[i+1] for i in range(max_coeff_idx)) and \
all(coeffs[i] >= coeffs[i+1] for i in range(max_coeff_idx, len(coeffs) - 1))Asked: 3 years ago
Seen: 499 times
Last updated: Aug 10 '22
Copyright Sage, 2010. Some rights reserved under creative commons license. Content on this site is licensed under a Creative Commons Attribution Share Alike 3.0 license.