Thue equation “funnel”

In this post I will be detailing a problem I have thought about for about a year, but has not made any progress on. The problem is one posed by my advisor, Professor Cameron Stewart, in the following paper:

C.L. Stewart, On the number of solutions of polynomial congruences and Thue equations, Journal of the AMS, (4) 4 (1991), 793-835.

His conjecture can be seen on page 816 of the above paper:

“… we conjecture that there exists an absolute constant c_0 such that for any binary form F \in \mathbb{Z}[x,y] with nonzero discriminant and degree at least three there exists a number C, which depends on F, such that if h is an integer larger than C then the Thue equation [F(x,y) = h] has at most c_0 solutions in coprime integers x and y.”

In the same paper, Stewart proves that the equation F(x,y) = h has no more than O((\deg F)^{1 + \omega(g)}) solutions, where g is a large divisor of h. The crucial point is that quite often, we can pick g which has few prime divisors, so that \omega(g) is small (see also this recent preprint due to Shabnam Akhtari:

However, this situation should not be typical. The basic idea is that when h is large with respect to the coefficients of F, that F(x,y) = h should behave `generically’, meaning that most of the points on the plane curve defined by the Thue equation should have few rational points. Indeed, generically the curve F(x,y) = h should be of general type (i.e. have genus at least 2), so one should not expect too many rational points by Faltings’ theorem.

Nevertheless, if one checks Stewart’s argument (this particular part is not necessarily new; Bombieri and Schmidt had used roughly the same idea in other papers), then one sees that a crucial ingredient is a p-adic reduction lemma. In particular, for a prime p and a positive integer k such that p^k || h, one can transform the equation F(x,y) = h into at most d = \deg F many equations of the form F_i(x,y) = hp^{-k}. Indeed, this is where the power of d appears in Stewart’s theorem. Another crucial input of Stewart, arguably the most novel part of his paper, is that he shows that the equation F(x,y) = n has few solutions provided that n is sufficiently small with respect to various invariants of F; in particular, he does not require that n = 1.

The deficit of Stewart’s argument, as with all other related results, is that it fundamentally obtains the ‘wrong’ bound: one should expect fewer solutions to the equation F(x,y) = h when h is large, not more. In particular, all arguments involve some sort of reduction or descent argument and reducing the equation to one where the right hand side is a small integer, thereby susceptible to various diophantine approximation arguments, at a cost that is not so severe. This type of argument is unlikely to be useful in resolving Stewart’s conjecture.

However, recent groundbreaking work due to Manjul Bhargava offer hope. In this paper, Bhargava shows that most hyperelliptic curves in the sense of some natural density with respect to the coefficients of the curve written as z^2 = f(x,y) with f a binary form with integral coefficients and even degree, have no rational points. One of the most ingenious constructions in the paper is an explicit representation of a binary form f(x,y) whenever the curve

z^2 = f(x,y)

has a rational point. Indeed, if the curve above has a rational point (z_0, x_0, y_0) say, then there exist integer matrices A,B \in \text{GL}_{2n}(\mathbb{Z}) with 2n = \deg f such that

f(x,y) = (-1)^n \det(Ax - By).

He then showed that most binary forms cannot be represented in the above way, thereby showing that most hyperelliptic curves have no rational points.

If one carries out Stewart’s p-adic reduction argument all the way down to 1, we see the following. Start with a generic equation F(x,y) = h, apply a descent argument, and end up with a bunch of equations of the form \mathcal{F}(x,y) = 1. In particular, the curve defined by

\mathcal{F}(x,y) = z^2

has a rational point! Therefore, it must be one of the rare binary forms with a representation of the form (-1)^n \det(Ax - By). I call this effect ‘funneling’, because we start with a generic, typical object and end up with a special object. Therefore, all of the initial data had to pass through a ‘general to special’ funnel. Since the final object is so rare, the initial object could not be too abundant; otherwise we would ‘violate’ Bhargava’s result.

Of course, the above paragraph is too coarse and fuzzy to formulate precise statements, much less proving a rigorous theorem. However, I still believe that this `funnel’ should exist and that it will lead to a solution to Stewart’s conjecture.

If you have any insights or ideas, please feel free to contact me, I would be extremely pleased if this question gets an answer.

On getting our names “whitesplained”

The phrase “mansplaining” is defined as follows, according to wikipedia: “to explain something to someone, typically a man to woman, in a manner regarded as condescending or patronizing.” It can easily be generalized to the case when a dominant social group tries to explain something to someone from a less dominant social group. This is particularly an issue when it comes to Chinese names and native English speakers trying to dictate to us how to say our own names ‘in English’.

This is particularly a problem for my wife, who has the very common Chinese surname ‘Wang’. According to this source (written in Chinese), there are over 93 million people in China (which does not include the diaspora, which my wife is a part of) with that surname. Most native English speakers pronounce it as spelt, which is like ‘bang’ but with a w in front. In fact, the correct pronunciation is closer to ‘Wong’ or ‘Wung’. There is no word in English that sounds exactly like how it sounds in Chinese, so one could be forgiven if they can’t pronounce it exactly right. This is not the problem.

The problem is that people tell my wife, when she corrects them on how to say her surname, that she’s incorrect. She hears people say “well in English it’s just said that way”. Many people tell her that it’s because she’s not a native speaker and therefore have an accent, that her pronunciation is flawed.

Why is this an issue? Well, it is absolutely insulting to say the least. To suggest that we don’t know how to say our own names is preposterous. The other issue at hand is that there is a flagrant lack of respect for our language. This issue flared up on a recent trip where we noticed a member of our group didn’t know the ‘correct’ pronunciation of Mexican food items such as quesadilla or tortilla. In particular, she didn’t know that the double l in these words are supposed to be a ‘yee’ sound. Upon closer examination, it seems that native English speakers have no problem adopting the actual correct pronunciation of a wide variety of foreign words. Nobody insists on pronouncing the following words ‘as written’, and in fact if you tried to pronounce these words using the usual rules of the English language, someone will correct you on it. These include:

Tortilla, quesadilla, fajita, gelato, latte, cappuccino, schadenfreude, de ja vu, jalapeño, etc.

Notice that all of these are words from another European language. It appears that native English speakers have no problem accepting that these words are not English words and therefore deserve to be pronounced the way that the original language conceived, but that Chinese surnames do not deserve the same courtesy. In fact, even on surnames, there is a double standard. If someone with a European sounding name insists on a pronunciation, nobody suggests that they simply need to say their name ‘the English way’.

So, today I say enough is enough. Please respect us when we say how our name is pronounced and do your best to pronounce it that way, instead of condescendingly imply that we’re speaking Chinglish.

What the Eurozone crisis teaches about us

As should be well known today, there is an ongoing financial crisis in the Eurozone centred around Greece. Namely, Greece defaulted on its IMF loan, and although a new bailout package has been more or less secured, the future of the Eurozone is now seriously questioned.

The Greek crisis teaches us many things here in Canada and the US. Although it is a less serious problem in Canada, Canada being much more egalitarian than our southern neighbours, we often hear about how great various socialist states in Europe are doing. Moreover, we distance ourselves from the kind of turmoil that is currently engulfing the Eurozone, as we think we are much more united than the different stated in Europe. However, I argue that we are closer to the latter than the former.

I believe the main obstacle to achieving more equitable social programs and wealth distribution schemes is our inability to identify all other citizens as a part of the same tribe. Indeed, being countries that depend on immigration for population growth and featuring a highly diverse population, racism inevitably comes into the picture. While a significant portion of Canadians and Americans are not white, they usually cannot be identified as ‘Canadian’ or ‘American’ unless they show their passports. This includes people who have lived on this continent for generations, and ironically, even includes native people. As such, the white dominated majority in both countries favour rules that keep the ‘others’ down. This is particularly severe in the US, where their prison system and the school-to-prison pipeline disproportionately affects minorities. This means that most people cannot accept a “we are all in this together” premise, applied to a national scale. Therefore, our innate racism and tendency to see people who look different as others prevent us from adopting socialist policies that we seem to like in Europe, because in those countries the population is almost uniformly homogeneous.

The Eurozone crisis is an exaggeration of what could happen if racial harmony deteriorates further. The root problem with the Eurozone is that while European leaders have a responsibility to safeguard the collective economic prosperity of the Eurozone, they are only held accountable for the prosperity of their own nation, as they are all nationally elected leaders (i.e. even if Angela Merkel does something that seriously damages the Eurozone but Germany comes out on top, making Merkel super unpopular in the Eurozone but popular in Germany, she will still be elected). This potential conflict of interest obstructs the European Union from becoming a functioning body. In a single nation, inevitably rich people and wealthy places subsidize poor people and impoverished places. This may not be done directly, but each nation collects taxes at the nation level, and have national social programs that directly or indirectly redistribute wealth. Wealthy places naturally pay more taxes, and in most cases benefit less proportionately from welfare programs, therefore creating a subsidy effect. Usually people don’t have too much of a problem with this because they need to accept, albeit begrudgingly perhaps, that their tax dollars are helping their fellow countrymen. This is not so on the scale of the EU, since Germans will likely never see Greeks as ‘their own’. This is further exacerbated by the strong national identities in Europe that have persisted for millennia.

While the problem is not quite so severe in Canada, some of our attitudes are just as problematic. For instance, the basic premise that paying taxes is a drain on one’s personal finances rather than an investment towards creating better public goods. Our sense of other prevents us from seeing the utility of ‘public goods’, because of the perception that ‘others’ will use them up more than us or even that ‘others’ will use these goods when they don’t ‘deserve to’. Only when we realize that investment in public goods leads to a better situation for all (and that it’s ok that this investment benefits the poor and disadvantaged more than those who are not poor and not disadvantaged), can we hope to create a better society.

My thoughts on reading the news

I read this article about a year ago and has contemplated on it from time to time over the year. However, some recent developments prompted me to think about the notion of reading the news a bit more.

From an early age, I was taught that reading the news daily (and not just local news, but national/international news) is a good habit. After neglecting this habit during my teens, I picked it up again in my early 20s and hasn’t stopped since. When I encountered the article linked above, I was dismissive at first, because how can there be an advantage to being under-informed? However, I was forced to go back on that thought during a recent news wave.

There are two big financial stories being told right now. One is centred in the Eurozone, and more specifically about Greece. I quickly realized how polarized the reporting is and how differently the story is told from different perspectives. On the one hand, the ‘right wing’ segments all focus on how lazy and deceitful the Greek people are and how they are basically robbing everyone else. On the other, the focus is on how austerity measures have crushed the Greek economy and that there is genuine suffering in Greece. However, the most important thing to take away, in line with the ‘focus on the car and forget the bridge’ analogy in the above article, is the lack of focus on the big picture, or exaggerating the situation.

Let us first discuss how the big picture is neglected. Greece is a tiny part of Europe. It accounts for less than 2% of the EU GDP, its population is a meagre 10 million, and it is geographically disconnected from the rest of the Eurozone. Its economic impact on the rest of the Eurozone, while not negligible perse, is not significant. So what is the big fuss about? It cannot be about the actual economic realities in Europe, because Greece is such a small part of it. One cannot suggest that the real economic impact will be catastrophic unless one is implying some sort of domino effect.

Next, how things are exaggerated. The biggest suggestion from the ashes of the Grexit debacle is that this might lead to the dissolution of the Eurozone (see here, for example). However, this conclusion seems odd, because it seems like the exact opposite happened. Despite tough rhetoric and strong defiance from the Greeks for weeks, at the last minute they caved, because they realized how important staying in the Eurozone is for their continued existence. Germany also folded in some sense, because they realized the fallout for being the black-hand in the death of the Eurozone would be disastrous for them. Therefore, the Eurozone is not really going anywhere because all parties from top and bottom have a vested interest in keeping it intact.

The next major instance of news distorting your perception rather than improving it is with China and its recent stock market crash. One can hardly find an objective analysis of what happened without the author injecting more of their own opinions on the matter. However, when one talks to someone actually on ground level, the situation seems not so bad. Most Chinese citizens are not involved directly with the stock market. In fact, even if claims that “millions” of Chinese people are buying stocks, in a country of 1.4 billion, that amounts to a lot less than 1% of the population. The true percentage is actually higher, which is an embarrassing testament to how a lot of news writers fail to comprehend basic arithmetic when trying to exaggerate (i.e. in their attempt to exaggerate the numbers they actually made the numbers underwhelming).

Next, the news is full of suggestions about the political ramifications for the Chinese government. Every time there is the slightest economic upheaval, suggestions abound that the end of the ‘Chinese miracle’ is near, and that the whole thing will come crashing down. This type of hysteria has existed for 30 years, since the ‘Chinese miracle’ began in the 1980s. This is the result of knee jerk reactions based on personal prejudices rather than a careful analysis. One has to remember that news writers are not scholars; their role is to write a good story that sells, rather than solid research that can hold up to peer review. Therefore, it is better to stick to sources closer to being the latter than the former.

Counter-intuitions of divergent series

A common mistake when it comes to understanding series, motivated by highschool-esque and contest mathematics type thought process, is to think that the behaviour of the series is dominated by the first (finitely many) terms. For example, we think of the series 1 + 2 + 3 + \cdots as ‘divergent’ because the first few terms seem to add up really quickly.

This intuition is challenged with the classic `smallest’ (note: there is no such thing as an actual smallest divergent series) divergent series: the harmonic series

\displaystyle \sum_{n=1}^\infty \frac{1}{n}.

It is challenging because it just `barely’ diverges. If we replace the exponent 1 of n by anything bigger, we will end up with a convergent series. Further, it doesn’t look divergent because the first few terms don’t add up very quickly.

The following really challenges our intuition: if we delete every positive integer n that contains the number 9 in their usual decimal representation, and then add up their reciprocals, we get something convergent! This seems ludicrous because of we try to add up the first 100 positive integers, sans those that need to be omitted, it seems to barely make a difference. However, if we think of the `later’ parts of the series, the argument becomes clear. For each positive integer n, let us consider the integers 10^n \leq k < 10^{n+1}. There are 8 admissible choices (namely 1,2,3,4,5,6,7,8) for the first digit, and 9 choices for each subsequent digit, for a total of 8 \cdot 9^n choices. Each denominator is of size at least 10^n, hence

\displaystyle \sum_{\substack{10^n \leq k < 10^{n+1} \\ k \text{ does not contain } 9 \text{ in its decimal representation}}} \frac{1}{k} < \frac{8 \cdot 9^n}{10^n}.

Let \sideset{}{'}\sum denote summation over positive integers that do not contain 9 in their decimal representation. Then, it follows that

\displaystyle \sideset{}{'} \sum_{m \geq 10} \frac{1}{m} \leq \sum_{n=1}^\infty \sideset{}{'} \sum_{10^n \leq k < 10^{n+1}} \frac{1}{k} \leq \sum_{n=1}^\infty \frac{8 \cdot 9^n}{10^n} < \infty.

Therefore, it is the tail of the series that determines its behaviour with respect to convergence, not the front part! This is a particularly vexing thing to emphasize to students who are trained to think in the fashion (and only in the fashion) “do a few computations, make a conjecture, then prove conjecture using contradiction or induction”.

“Fuel” and “Engines” in mathematics

Note: First blog post in a while, been a crazy few months.

There are many analogies mathematicians use to try to explain the abstract realm they work in to laymen of various levels. For example, Freeman Dyson had tried to categorize mathematicians as “frogs” or “birds” and Tim Gowers tried to categorize them as “problem solvers” or “theory crafters”. In a post on MathOverflow, I tried to categorize certain theorems as ‘workhorses’.

Another useful way I recently thought of to think about various theorems and technique in mathematics is to think of them as “fuel” or “engines”. Fuel and engines are both parts of “machines”, which comprise theories and programs in mathematics. Fuel is very versatile; an improvement in fuel will almost certainly mean that all engines which can use said fuel can operate better. Engines are more specific; they are designed to perform a specific function. A car engine for example is useless if you try to make it chop vegetables, unless you design an entire system which chops vegetables while being powered by a car engine.

This is a good way to explain just how the approaches Yitang Zhang and James Maynard differ. One can think of the original sieve mechanism devised by Goldston, Pintz, and Yildirim, now known as the GPY-sieve. In a sense they constructed a ‘machine’ which allows us to turn churn out small gaps between primes. The engine of the machine is the GPY sieve mechanism; while the fuel (or the raw input) is the Bombieri-Vinogradov theorem. The machine works quite efficiently in that it produces gaps that are arbitrarily small compared to the average gap; but the engine just can’t put out enough power to establish bounded gaps between primes with the current quality of the fuel. This situation was thought to be as far as one can go back in 2013.

Yitang Zhang came along and showed that while it is not possible to develop an improved ‘all purpose’ fuel, i.e. an improvement on the level of distribution present in the Bombieri-Vinogradov theorem, one can devise a specialized and more efficient fuel that works specifically for the GPY engine. This extra boost in power was sufficient to establish bounded gaps.

Maynard, however, showed that one can rip out the GPY engine and replace it with a much more efficient and powerful engine that requires much less fuel and can output much more power. This is Maynard’s multi-dimensional sieve; something thought to be impossible previously.

The problem is that Maynard’s engine doesn’t run on Zhang’s fuel; because the latter is specialized for the GYP engine. Modifying the Maynard engine so that it runs on the Zhang fuel would be the next step forward towards the bounded gap problem.