https://mathoverflow.net/questions/470951/is-the-largest-root-of-a-random-polynomial-more-likely-to-be-real-than-complex Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange [ ] Loading... 1. + Tour Start here for a quick overview of the site + Help Center Detailed answers to any questions you might have + Meta Discuss the workings and policies of this site + About Us Learn more about Stack Overflow the company, and our products 2. 3. current community + MathOverflow help chat + MathOverflow Meta your communities Sign up or log in to customize your list. more stack exchange communities company blog 4. 5. Log in 6. Sign up MathOverflow 1. 1. Home 2. Questions 3. Tags 4. 5. Users 6. Unanswered Is the largest root of a random polynomial more likely to be real than complex? Ask Question Asked today Modified today Viewed 14k times 45 $\begingroup$ This question might be hard because it got $35$ upvotes in MSE and also had a $200$ points bounty by Jyrki Lahtonen but it was unanswered. So I am posting it in MO. The number of real roots of a random polynomial with real coefficients is much smaller than the number of complex roots. Assume that the coefficients are independently and uniformly random in $ (-1,1)$, for if not then we can divide each coefficient by the coefficient with the largest absolute value to scale each coefficient to $(-1,1)$. The number of real roots of a polynomial of degree $n$ is asymptotic to $\displaystyle \frac{2\log n}{\pi} + o(1)$. This means that the number of complex roots is approximately $\ displaystyle n - \frac{2\log n}{\pi}$. Similar asymptotics hold for other distribution of the coefficients. **Definition **: The largest (or smallest) root of a polynomial is the root with the largest (or smallest) modulus. Roots scatter plot The above graph shows the roots of one such polynomial with degree $101$. The largest root is in the top right corner in green. We can ask if the largest (or the smallest) root is more likely to be real or complex? Since there are exponentially more complex roots than real roots as seen from the above asymptotic, my naive guess was that the largest (or the smallest) root is more likely to be complex. However experimental data proved to be quite counterintuitive. Probability that the largest root is real The data show that 1. Probability that the largest (or smallest) root is real is greater than the probability that it is complex. 2. And this probability decreases to some value near $1/2$ as $n \to \infty$ as shown in the above graph (created using a Monte Carlo simulation with $10^5$ trials for each value of $n$). 3. Note: Instead of uniform distribution, if we assume that the coefficients are normally distributed with mean $0$ and standard deviation $1$ and scaled to $(-1,1)$, the above observation and limiting probabilities hold. It is counterintuitive that, despite being much (exponentially) fewer in number, real roots are more likely to contain both the largest and the smallest roots of a random polynomial. In this sense, the largest and the smallest roots are both biased towards reals. Question 1: What is the reason for this bias? Question 2: Does the probability that the largest (or the smallest) root of a polynomial of degree $n$ is real converge (to some value near $\frac{1}{2}$ as $n \to \infty$)? Note: We can quantify the observed bias as follows. Let $P(L|R)$ be the probability that a root is the largest given that it is real and let $P(L|C)$ be the probability that a root is the largest given that it is complex. Similarly, let $P(S|R)$ be the probability that a root is the smallest given that it is real and let $P(S|C)$ be the probability that a root is the smallest given that it is complex. Then the experimental data say that $$ P(L|R) = P(S|R) \approx \frac{\pi}{4\log n}, $$ $$ P(L|C) = P(S|C) \approx \frac{\pi}{2n\pi - 4\log n}. $$ Update: In the linked MSE post, it has now been proved that the probability that the largest root is real is at least $$ \frac{23-16\sqrt{2}}{6} \approx 6.2 \% $$ Related: What is the probability that the absolute value of the roots of a polynomial of degree $n$ is greater than $x$? * pr.probability * real-analysis * polynomials Share Cite Improve this question Follow edited 26 mins ago J. W. Tanner's user avatar J. W. Tanner 64544 silver badges1212 bronze badges asked 18 hours ago Nilotpal Kanti Sinha's user avatar Nilotpal Kanti SinhaNilotpal Kanti Sinha 4,99022 gold badges3030 silver badges4747 bronze badges $\endgroup$ 8 * 1 $\begingroup$ Nice question! However, it does not look like the probability is converging to $1/2$ but rather to something strictly above $1/2$, if it converges at all. $\endgroup$ - mathworker21 17 hours ago * 1 $\begingroup$ @mathworker21 Valid point; looks somewhere about $0.51$. I have updated the post. Running some computations for larger value of $n$ to see where is it converging to is at all. $ \endgroup$ - Nilotpal Kanti Sinha 16 hours ago * 1 $\begingroup$ The above graph shows also an example where the smallest root is real. If you would not have this green outlier dot, also the largest root would be real. Does the root scatter plot always looks similar to that? Then it seems reasonable that the smallest/largest roots are real. $\endgroup$ - Albert 12 hours ago * 3 $\begingroup$ Just an idea, it might be profitable to think of your polynomial as the characteristic polynomial of a random matrix: there is a vast literature on the eigenvalues of such matrices, for instance see a paper on a related (but different) question by Edelman: Edelman, Alan. "The probability that a random real gaussian matrix has real eigenvalues, related distributions, and the circular law." journal of multivariate analysis 60.2 (1997): 203-232. $\endgroup$ - B. W. Lewis 10 hours ago * $\begingroup$ Fussy quibble: "The largest (or smallest) root of a polynomial is the root with the largest (or smallest) modulus." should be "A largest (smallest) root of a polynomial is a root with the largest (smallest) modulus." Generically, smallest/ largest roots come in complex conjugate pairs, both having the same modulus. $\endgroup$ - Eric Towers 8 hours ago | Show 3 more comments 0 Sorted by: Reset to default [Highest score (default) ] Know someone who can answer? Share a link to this question via email, Twitter, or Facebook. Your Answer [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] Thanks for contributing an answer to MathOverflow! * Please be sure to answer the question. Provide details and share your research! But avoid ... * Asking for help, clarification, or responding to other answers. * Making statements based on opinion; back them up with references or personal experience. Use MathJax to format equations. MathJax reference. To learn more, see our tips on writing great answers. Draft saved Draft discarded [ ] Sign up or log in Sign up using Google Sign up using Facebook Sign up using Email and Password Submit Post as a guest Name [ ] Email Required, but never shown [ ] Post as a guest Name [ ] Email Required, but never shown [ ] Post Your Answer Discard By clicking "Post Your Answer", you agree to our terms of service and acknowledge you have read our privacy policy. Browse other questions tagged * pr.probability * real-analysis * polynomials or ask your own question. * Featured on Meta * Imgur image URL migration: Coming soon to a Stack Exchange site near you! * Our Partnership with OpenAI * Testing a new version of Stack Overflow Jobs Related 4 A known Lemma on the largest root of a polynomial and its derivatives? 1 k-th largest root in common interlacing polynomials 11 Variance of the roots of a complex polynomial 15 Random Diophantine polynomials: Percent solvable? 4 counting complex roots which are root of unity times a real number 1 Bounds on the smallest real positive root of a polynomial 11 A question on the real root of a polynomial 2 Expected fraction of roots in the unit disc of random polynomial with Gaussian coefficients Question feed Subscribe to RSS Question feed To subscribe to this RSS feed, copy and paste this URL into your RSS reader. [https://mathoverflow] * MathOverflow * Tour * Help * Chat * Contact * Feedback Company * Stack Overflow * Teams * Advertising * Collectives * Talent * About * Press * Legal * Privacy Policy * Terms of Service * Cookie Settings * Cookie Policy Stack Exchange Network * Technology * Culture & recreation * Life & arts * Science * Professional * Business * API * Data * Blog * Facebook * Twitter * LinkedIn * Instagram Site design / logo (c) 2024 Stack Exchange Inc; user contributions licensed under CC BY-SA. rev 2024.5.10.8902