War's Human Cost
War's Human Cost
I didn't start crying until I was outside a McDonald's in an Olempin, Poland rest area on highway S17.
Children pick toys at a refugee center, Olempin, Poland, March 4, 2022.
Refugee children, mostly alone with their mothers, but occasionally with a gray-haired grandfather or non-Ukrainian father, were coaxed into picking a toy from boxes provided by a kind-hearted company and volunteers.
I went to Warsaw to continue my research on my family's history during the Holocaust. In light of the ongoing Ukrainian conflict, I asked former colleagues in the US Department of Defense and Intelligence Community if it was safe to travel there. They said yes, as Poland was a NATO member.
I stayed in a hotel in the Warsaw Ghetto, where 90% of my mother's family was murdered in the Holocaust. Across the street was the first Warsaw Judenrat. It was two blocks away from the apartment building my mother's family had owned and lived in, now dilapidated and empty.
Building of my great-grandfather, December 2021.
A mass grave of thousands of rocks for those killed in the Warsaw Ghetto, I didn't cry when I touched its cold walls.
Warsaw Jewish Cemetery, 200,000–300,000 graves.
Mass grave, Warsaw Jewish Cemetery.
My mother's family had two homes, one in Warszawa and the rural one was a forest and sawmill complex in Western Ukraine. For the past half-year, a local Ukrainian historian had been helping me discover faint traces of her family’s life there — in fact, he had found some people still alive who remembered the sawmill and that it belonged to my mother’s grandfather. The historian was good at his job, and we had become close.
My historian friend, December 2021, talking to a Ukrainian.
With war raging, my second trip to Warsaw took on a different mission. To see his daughter and one-year-old grandson, I drove east instead of to Ukraine. They had crossed the border shortly after the war began, leaving men behind, and were now staying with a friend on Poland's eastern border.
I entered after walking up to the house and settling with the dog. The grandson greeted me with a huge smile and the Ukrainian word for “daddy,” “Tato!” But it was clear he was awaiting his real father's arrival, and any man he met would be so tentatively named.
After a few moments, the boy realized I was only a stranger. He had musical talent, like his mother and grandfather, both piano teachers, as he danced to YouTube videos of American children's songs dubbed in Ukrainian, picking the ones he liked and crying when he didn't.
Songs chosen by my historian friend's grandson, March 4, 2022
He had enough music and began crying regardless of the song. His mother picked him up and started nursing him, saying she was worried about him. She had no idea where she would live or how she would survive outside Ukraine. She showed me her father's family history of losses in the Holocaust, which matched my own research.
After an hour of drinking tea and trying to speak of hope, I left for the 3.5-hour drive west to Warsaw.
It was unlike my drive east. It was reminiscent of the household goods-filled carts pulled by horses and people fleeing war 80 years ago.
Jewish refugees relocating, USHMM Holocaust Encyclopaedia, 1939.
The carefully chosen trinkets by children to distract them from awareness of what is really happening and the anxiety of what lies ahead, made me cry despite all my research on the Holocaust. There is no way for them to communicate with their mothers, who are worried, absent, and without their fathers.
It's easy to see war as a contest of nations' armies, weapons, and land. The most costly aspect of war is its psychological toll. My father screamed in his sleep from nightmares of his own adolescent trauma in Warsaw 80 years ago.
Survivor father studying engineering, 1961.
In the airport, I waited to return home while Ukrainian public address systems announced refugee assistance. Like at McDonald's, many mothers were alone with their children, waiting for a flight to distant relatives.
That's when I had my worst trip experience.
A woman near me, clearly a refugee, answered her phone, cried out, and began wailing.
The human cost of war descended like a hammer, and I realized that while I was going home, she never would
More on Current Events
Blake Montgomery
3 years ago
Explaining Twitter Files
Elon Musk, Matt Taibbi, the 'Twitter Files,' and Hunter Biden's laptop: what gives?
Explaining Twitter Files
Matt Taibbi released "The Twitter Files," a batch of emails sent by Twitter executives discussing the company's decision to stop an October 2020 New York Post story online.
What's on Twitter? New York Post and Fox News call them "bombshell" documents. Or, as a Post columnist admitted, are they "not the smoking gun"? Onward!
What started this?
The New York Post published an exclusive, potentially explosive story in October 2020: Biden's Secret Emails: Ukrainian executive thanks Hunter Biden for'meeting' veep dad. The story purported to report the contents of a laptop brought to the tabloid by a Delaware computer repair shop owner who said it belonged to President Biden's second son, Hunter Biden. Emails and files on the laptop allegedly showed how Hunter peddled influence with Ukranian businessmen and included a "raunchy 12-minute video" of Hunter smoking crack and having sex.
Twitter banned links to the Post story after it was published, calling it "hacked material." The Post's Twitter account was suspended for multiple days.
Why? Yoel Roth, Twitter's former head of trust and safety, said the company couldn't verify the story, implying they didn't trust the Post.
Twitter's stated purpose rarely includes verifying news stories. This seemed like intentional political interference. This story was hard to verify because the people who claimed to have found the laptop wouldn't give it to other newspapers. (Much of the story, including Hunter's business dealings in Ukraine and China, was later confirmed.)
Roth: "It looked like a hack and leak."
So what are the “Twitter Files?”
Twitter's decision to bury the story became a political scandal, and new CEO Elon Musk promised an explanation. The Twitter Files, named after Facebook leaks.
Musk promised exclusive details of "what really happened" with Hunter Biden late Friday afternoon. The tweet was punctuated with a popcorn emoji.
Explaining Twitter Files
Three hours later, journalist Matt Taibbi tweeted more than three dozen tweets based on internal Twitter documents that revealed "a Frankensteinian tale of a human-built mechanism grown out of its designer's control."
Musk sees this release as a way to shape Twitter's public perception and internal culture in his image. We don't know if the CEO gave Taibbi the documents. Musk hyped the document dump before and during publication, but Taibbi cited "internal sources."
Taibbi shares email screenshots showing Twitter execs discussing the Post story and blocking its distribution. Taibbi says the emails show Twitter's "extraordinary steps" to bury the story.
Twitter communications chief Brandon Borrman has the most damning quote in the Files. Can we say this is policy? The story seemed unbelievable. It seemed like a hack... or not? Could Twitter, which ex-CEO Dick Costolo called "the free speech wing of the free speech party," censor a news story?
Many on the right say the Twitter Files prove the company acted at the behest of Democrats. Both parties had these tools, writes Taibbi. In 2020, both the Trump White House and Biden campaign made requests. He says the system for reporting tweets for deletion is unbalanced because Twitter employees' political donations favor Democrats. Perhaps. These donations may have helped Democrats connect with Twitter staff, but it's also possible they didn't. No emails in Taibbi's cache show these alleged illicit relations or any actions Twitter employees took as a result.
Even Musk's supporters were surprised by the drop. Miranda Devine of the New York Post told Tucker Carlson the documents weren't "the smoking gun we'd hoped for." Sebastian Gorka said on Truth Social, "So far, I'm deeply underwhelmed." DC Democrats collude with Palo Alto Democrats. Whoop!” The Washington Free Beacon's Joe Simonson said the Twitter files are "underwhelming." Twitter was staffed by Democrats who did their bidding. (Why?)
If "The Twitter Files" matter, why?
These emails led Twitter to suppress the Hunter Biden laptop story has real news value. It's rare for a large and valuable company like Twitter to address wrongdoing so thoroughly. Emails resemble FOIA documents. They describe internal drama at a company with government-level power. Katie Notopoulos tweeted, "Any news outlet would've loved this scoop!" It's not a'scandal' as teased."
Twitter's new owner calls it "the de facto public town square," implying public accountability. Like a government agency. Though it's exciting to receive once-hidden documents in response to a FOIA, they may be boring and tell you nothing new. Like Twitter files. We learned how Twitter blocked the Post's story, but not why. Before these documents were released, we knew Twitter had suppressed the story and who was involved.
These people were disciplined and left Twitter. Musk fired Vijaya Gadde, the former CLO who reportedly played a "key role" in the decision. Roth quit over Musk's "dictatorship." Musk arrived after Borrman left. Jack Dorsey, then-CEO, has left. Did those who digitally quarantined the Post's story favor Joe Biden and the Democrats? Republican Party opposition and Trump hatred? New York Post distaste? According to our documents, no. Was there political and press interference? True. We knew.
Taibbi interviewed anonymous ex-Twitter employees about the decision; all expressed shock and outrage. One source said, "Everyone knew this was fucked." Since Taibbi doesn't quote that expletive, we can assume the leaked emails contained few or no sensational quotes. These executives said little to support nefarious claims.
Outlets more invested in the Hunter Biden story than Gizmodo seem vexed by the release and muted headlines. The New York Post, which has never shied away from a blaring headline in its 221-year history, owns the story of Hunter Biden's laptop. Two Friday-night Post alerts about Musk's actions were restrained. Elon Musk will drop Twitter files on NY Post-Hunter Biden laptop censorship today. Elon Musk's Twitter dropped Post censorship details from Biden's laptop. Fox News' Apple News push alert read, "Elon Musk drops Twitter censorship documents."
Bombshell, bombshell, bombshell… what, exactly, is the bombshell? Maybe we've heard this story too much and are missing the big picture. Maybe these documents detail a well-documented decision.
The Post explains why on its website. "Hunter Biden laptop bombshell: Twitter invented reason to censor Post's reporting," its headline says.
Twitter's ad hoc decision to moderate a tabloid's content is not surprising. The social network had done this for years as it battled toxic users—violent white nationalists, virulent transphobes, harassers and bullies of all political stripes, etc. No matter how much Musk crows, the company never had content moderation under control. Buzzfeed's 2016 investigation showed how Twitter has struggled with abusive posters since 2006. Jack Dorsey and his executives improvised, like Musk.
Did the US government interfere with the ex-social VP's media company? That's shocking, a bombshell. Musk said Friday, "Twitter suppressing free speech by itself is not a 1st amendment violation, but acting under government orders with no judicial review is." Indeed! Taibbi believed this. August 2022: "The laptop is secondary." Zeynep Tufecki, a Columbia professor and New York Times columnist, says the FBI is cutting true story distribution. Taibbi retracted the claim Friday night: "I've seen no evidence of government involvement in the laptop story."
What’s the bottom line?
I'm still not sure what's at stake in the Hunter Biden scandal after dozens of New York Post articles, hundreds of hours of Fox News airtime, and thousands of tweets. Briefly: Joe Biden's son left his laptop with a questionable repairman. FBI confiscated it? The repairman made a copy and gave it to Rudy Giuliani's lawyer. The Post got it from Steve Bannon. On that laptop were videos of Hunter Biden smoking crack, cavorting with prostitutes, and emails about introducing his father to a Ukrainian businessman for $50,000 a month. Joe Biden urged Ukraine to fire a prosecutor investigating the company. What? The story seems to be about Biden family business dealings, right?
The discussion has moved past that point anyway. Now, the story is the censorship of it. Adrienne Rich wrote in "Diving Into the Wreck" that she came for "the wreck and not the story of the wreck" No matter how far we go, Hunter Biden's laptop is done. Now, the crash's story matters.
I'm dizzy. Katherine Miller of BuzzFeed wrote, "I know who I believe, and you probably do, too. To believe one is to disbelieve the other, which implicates us in the decision; we're stuck." I'm stuck. Hunter Biden's laptop is a political fabrication. You choose. I've decided.
This could change. Twitter Files drama continues. Taibbi said, "Much more to come." I'm dizzy.
Isaiah McCall
3 years ago
There is a new global currency emerging, but it is not bitcoin.
America should avoid BRICS
Vladimir Putin has watched videos of Muammar Gaddafi's CIA-backed demise.
Gaddafi...
Thief.
Did you know Gaddafi wanted a gold-backed dinar for Africa? Because he considered our global financial system was a Ponzi scheme, he wanted to discontinue trading oil in US dollars.
Or, Gaddafi's Libya enjoyed Africa's highest quality of living before becoming freed. Pictured:
Vladimir Putin is a nasty guy, but he had his reasons for not mentioning NATO assisting Ukraine in resisting US imperialism. Nobody tells you. Sure.
The US dollar's corruption post-2008, debasement by quantitative easing, and lack of value are key factors. BRICS will replace the dollar.
BRICS aren't bricks.
Economy-related.
Brazil, Russia, India, China, and South Africa have cooperated for 14 years to fight U.S. hegemony with a new international currency: BRICS.
BRICS is mostly comical. Now. Saudi Arabia, the second-largest oil hegemon, wants to join.
So what?
The New World Currency is BRICS
Russia was kicked out of G8 for its aggressiveness in Crimea in 2014.
It's now G7.
No biggie, said Putin, he said, and I quote, “Bon appetite.”
He was prepared. China, India, and Brazil lead the New World Order.
Together, they constitute 40% of the world's population and, according to the IMF, 50% of the world's GDP by 2030.
Here’s what the BRICS president Marcos Prado Troyjo had to say earlier this year about no longer needing the US dollar: “We have implemented the mechanism of mutual settlements in rubles and rupees, and there is no need for our countries to use the dollar in mutual settlements. And today a similar mechanism of mutual settlements in rubles and yuan is being developed by China.”
Ick. That's D.C. and NYC warmongers licking their chops for WW3 nasty.
Here's a lovely picture of BRICS to relax you:
If Saudi Arabia joins BRICS, as President Mohammed Bin Salman has expressed interest, a majority of the Middle East will have joined forces to construct a new world order not based on the US currency.
I'm not sure of the new acronym.
SBRICSS? CIRBSS? CRIBSS?
The Reason America Is Harvesting What It Sowed
BRICS began 14 years ago.
14 years ago, what occurred? Concentrate. It involved CDOs, bad subprime mortgages, and Wall Street quants crunching numbers.
2008 recession
When two nations trade, they do so in US dollars, not Euros or gold.
What happened when 2008, an avoidable crisis caused by US banks' cupidity and ignorance, what happened?
Everyone WORLDWIDE felt the pain.
Mostly due to corporate America's avarice.
This should have been a warning that China and Russia had enough of our bs. Like when France sent a battleship to America after Nixon scrapped the gold standard. The US was warned to shape up or be dethroned (or at least try).
Nixon improved in 1971. Kinda. Invented PetroDollar.
Another BS system that unfairly favors America and possibly pushed Russia, China, and Saudi Arabia into BRICS.
The PetroDollar forces oil-exporting nations to trade in US dollars and invest in US Treasury bonds. Brilliant. Genius evil.
Our misdeeds are:
In conflicts that are not its concern, the USA uses the global reserve currency as a weapon.
Targeted nations abandon the dollar, and rightfully so, as do nations that depend on them for trade in vital resources.
The dollar's position as the world's reserve currency is in jeopardy, which could have disastrous economic effects.
Although we have actually sown our own doom, we appear astonished. According to the Bible, whomever sows to appease his sinful nature will reap destruction from that nature whereas whoever sows to appease the Spirit will reap eternal life from the Spirit.
Americans, even our leaders, lack caution and delayed pleasure. When our unsustainable systems fail, we double down. Bailouts of the banks in 2008 were myopic, puerile, and another nail in America's hegemony.
America has screwed everyone.
We're unpopular.
The BRICS's future
It's happened before.
Saddam Hussein sold oil in Euros in 2000, and the US invaded Iraq a month later. The media has devalued the word conspiracy. The Iraq conspiracy.
There were no WMDs, but NYT journalists like Judy Miller drove Americans into a warmongering frenzy because Saddam would ruin the PetroDollar. Does anyone recall that this war spawned ISIS?
I think America has done good for the world. You can make a convincing case that we're many people's villain.
Learn more in Confessions of an Economic Hitman, The Devil's Chessboard, or Tyranny of the Federal Reserve. Or ignore it. That's easier.
We, America, should extend an olive branch, ask for forgiveness, and learn from our faults, as the Tao Te Ching advises. Unlikely. Our population is apathetic and stupid, and our government is corrupt.
Argentina, Iran, Egypt, and Turkey have also indicated interest in joining BRICS. They're also considering making it gold-backed, making it a new world reserve currency.
You should pay attention.
Thanks for reading!
Cory Doctorow
3 years ago
The downfall of the Big Four accounting companies is just one (more) controversy away.
Economic mutual destruction.
Multibillion-dollar corporations never bothered with an independent audit, and they all lied about their balance sheets.
It's easy to forget that the Big Four accounting firms are lousy fraud enablers. Just because they sign off on your books doesn't mean you're not a hoax waiting to erupt.
This is *crazy* Capitalism depends on independent auditors. Rich folks need to know their financial advisers aren't lying. Rich folks usually succeed.
No accounting. EY, KPMG, PWC, and Deloitte make more money consulting firms than signing off on their accounts.
The Big Four sign off on phony books because failing to make friends with unscrupulous corporations may cost them consulting contracts.
The Big Four are the only firms big enough to oversee bankruptcy when they sign off on fraudulent books, as they did for Carillion in 2018. All four profited from Carillion's bankruptcy.
The Big Four are corrupt without any consequences for misconduct. Who can forget when KPMG's top management was fined millions for helping auditors cheat on ethics exams?
Consulting and auditing conflict. Consultants help a firm cover its evil activities, such as tax fraud or wage theft, whereas auditors add clarity to a company's finances. The Big Four make more money from cooking books than from uncooking them, thus they are constantly embroiled in scandals.
If a major scandal breaks, it may bring down the entire sector and substantial parts of the economy. Jim Peterson explains system risk for The Dig.
The Big Four are voluntary private partnerships where accountants invest their time, reputations, and money. If a controversy threatens the business, partners who depart may avoid scandal and financial disaster.
When disaster looms, each partner should bolt for the door, even if a disciplined stay-and-hold posture could weather the storm. This happened to Arthur Andersen during Enron's collapse, and a 2006 EU report recognized the risk to other corporations.
Each partner at a huge firm knows how much dirty laundry they've buried in the company's garden, and they have well-founded suspicions about what other partners have buried, too. When someone digs, everyone runs.
If a firm confronts substantial litigation damages or enforcement penalties, it could trigger the collapse of one of the Big Four. That would be bad news for the firm's clients, who would have trouble finding another big auditor.
Most of the world's auditing capacity is concentrated in four enormous, brittle, opaque, compromised organizations. If one of them goes bankrupt, the other three won't be able to take on its clients.
Peterson: Another collapse would strand many of the world's large public businesses, leaving them unable to obtain audit views for their securities listings and regulatory compliance.
Count Down: The Past, Present, and Uncertain Future of the Big Four Accounting Firms is in its second edition.
https://www.emerald.com/insight/publication/doi/10.1108/9781787147003
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Jack Shepherd
3 years ago
A Dog's Guide to Every Type of Zoom Call Participant
Are you one of these Zoom dogs?
The Person Who Is Apparently Always on Mute
Waffles thinks he can overpower the mute button by shouting loudly.
The person who believed their camera to be off
Barkley's used to remote work, but he hasn't mastered the "Stop Video" button. Everyone is affected.
Who is driving for some reason, exactly?
Why is Pumpkin always late? Who knows? Shouldn't she be driving? If you could hear her over the freeway, she'd answer these questions.
The Person With the Amazing Bookcase
Cicero likes to use SAT-words like "leverage" and "robust" in Zoom sessions, presumably from all the books he wants you to see behind him.
The Individual Who Is Unnecessarily Dressed
We hope Bandit is going somewhere beautiful after this meeting, or else he neglected the quarterly earnings report and is overcompensating to distract us.
The person who works through lunch in between zoom calls
Barksworth has back-to-back meetings all day, so you can watch her eat while she talks.
The Person Who Is A Little Too Comfy
Hercules thinks Zoom meetings happen between sleeps. He'd appreciate everyone speaking more quietly.
The Person Who Answered the Phone Outside
Frisbee has a gorgeous backyard and lives in a place with great weather year-round, and she wants you to think about that during the daily team huddle.
Who Wants You to Pay Attention to Their Pet
Snickers hasn't listened to you in 20 minutes unless you tell her how cute her kitten is.
One who is, for some reason, positioned incorrectly on the screen
Nelson's meetings consist primarily of attempting to figure out how he positioned his laptop so absurdly.
The person who says too many goodbyes
Zeus waves farewell like it's your first day of school while everyone else searches for the "Leave Meeting" button. It's nice.
He who has a poor internet connection
Ziggy's connectivity problems continue... She gives a long speech as everyone waits awkwardly to inform her they missed it.
The Clearly Multitasking Person
Tinkerbell can play fetch during the monthly staff meeting if she works from home, but that's not a good idea.
The Person Using Zoom as a Makeup and Hair Mirror
If Gail and Bob knew Zoom had a "hide self view" option, they'd be distraught.
The person who feels at ease with simply leaving
Rusty bails when a Zoom conference is over. Rusty's concept is decent.
CNET
4 years ago
How a $300K Bored Ape Yacht Club NFT was accidentally sold for $3K
The Bored Ape Yacht Club is one of the most prestigious NFT collections in the world. A collection of 10,000 NFTs, each depicting an ape with different traits and visual attributes, Jimmy Fallon, Steph Curry and Post Malone are among their star-studded owners. Right now the price of entry is 52 ether, or $210,000.
Which is why it's so painful to see that someone accidentally sold their Bored Ape NFT for $3,066.
Unusual trades are often a sign of funny business, as in the case of the person who spent $530 million to buy an NFT from themselves. In Saturday's case, the cause was a simple, devastating "fat-finger error." That's when people make a trade online for the wrong thing, or for the wrong amount. Here the owner, real name Max or username maxnaut, meant to list his Bored Ape for 75 ether, or around $300,000. Instead he accidentally listed it for 0.75. One hundredth the intended price.
It was bought instantaneously. The buyer paid an extra $34,000 to speed up the transaction, ensuring no one could snap it up before them. The Bored Ape was then promptly listed for $248,000. The transaction appears to have been done by a bot, which can be coded to immediately buy NFTs listed below a certain price on behalf of their owners in order to take advantage of these exact situations.
"How'd it happen? A lapse of concentration I guess," Max told me. "I list a lot of items every day and just wasn't paying attention properly. I instantly saw the error as my finger clicked the mouse but a bot sent a transaction with over 8 eth [$34,000] of gas fees so it was instantly sniped before I could click cancel, and just like that, $250k was gone."
"And here within the beauty of the Blockchain you can see that it is both honest and unforgiving," he added.
Fat finger trades happen sporadically in traditional finance -- like the Japanese trader who almost bought 57% of Toyota's stock in 2014 -- but most financial institutions will stop those transactions if alerted quickly enough. Since cryptocurrency and NFTs are designed to be decentralized, you essentially have to rely on the goodwill of the buyer to reverse the transaction.
Fat finger errors in cryptocurrency trades have made many a headline over the past few years. Back in 2019, the company behind Tether, a cryptocurrency pegged to the US dollar, nearly doubled its own coin supply when it accidentally created $5 billion-worth of new coins. In March, BlockFi meant to send 700 Gemini Dollars to a set of customers, worth roughly $1 each, but mistakenly sent out millions of dollars worth of bitcoin instead. Last month a company erroneously paid a $24 million fee on a $100,000 transaction.
Similar incidents are increasingly being seen in NFTs, now that many collections have accumulated in market value over the past year. Last month someone tried selling a CryptoPunk NFT for $19 million, but accidentally listed it for $19,000 instead. Back in August, someone fat finger listed their Bored Ape for $26,000, an error that someone else immediately capitalized on. The original owner offered $50,000 to the buyer to return the Bored Ape -- but instead the opportunistic buyer sold it for the then-market price of $150,000.
"The industry is so new, bad things are going to happen whether it's your fault or the tech," Max said. "Once you no longer have control of the outcome, forget and move on."
The Bored Ape Yacht Club launched back in April 2021, with 10,000 NFTs being sold for 0.08 ether each -- about $190 at the time. While NFTs are often associated with individual digital art pieces, collections like the Bored Ape Yacht Club, which allow owners to flaunt their NFTs by using them as profile pictures on social media, are becoming increasingly prevalent. The Bored Ape Yacht Club has since become the second biggest NFT collection in the world, second only to CryptoPunks, which launched in 2017 and is considered the "original" NFT collection.
Vitalik
4 years ago
An approximate introduction to how zk-SNARKs are possible (part 2)
If tasked with the problem of coming up with a zk-SNARK protocol, many people would make their way to this point and then get stuck and give up. How can a verifier possibly check every single piece of the computation, without looking at each piece of the computation individually? But it turns out that there is a clever solution.
Polynomials
Polynomials are a special class of algebraic expressions of the form:
- x+5
- x^4
- x^3+3x^2+3x+1
- 628x^{271}+318x^{270}+530x^{269}+…+69x+381
i.e. they are a sum of any (finite!) number of terms of the form cx^k
There are many things that are fascinating about polynomials. But here we are going to zoom in on a particular one: polynomials are a single mathematical object that can contain an unbounded amount of information (think of them as a list of integers and this is obvious). The fourth example above contained 816 digits of tau, and one can easily imagine a polynomial that contains far more.
Furthermore, a single equation between polynomials can represent an unbounded number of equations between numbers. For example, consider the equation A(x)+ B(x) = C(x). If this equation is true, then it's also true that:
- A(0)+B(0)=C(0)
- A(1)+B(1)=C(1)
- A(2)+B(2)=C(2)
- A(3)+B(3)=C(3)
And so on for every possible coordinate. You can even construct polynomials to deliberately represent sets of numbers so you can check many equations all at once. For example, suppose that you wanted to check:
- 12+1=13
- 10+8=18
- 15+8=23
- 15+13=28
You can use a procedure called Lagrange interpolation to construct polynomials A(x) that give (12,10,15,15) as outputs at some specific set of coordinates (eg. (0,1,2,3)), B(x) the outputs (1,8,8,13) on thos same coordinates, and so forth. In fact, here are the polynomials:
- A(x)=-2x^3+\frac{19}{2}x^2-\frac{19}{2}x+12
- B(x)=2x^3-\frac{19}{2}x^2+\frac{29}{2}x+1
- C(x)=5x+13
Checking the equation A(x)+B(x)=C(x) with these polynomials checks all four above equations at the same time.
Comparing a polynomial to itself
You can even check relationships between a large number of adjacent evaluations of the same polynomial using a simple polynomial equation. This is slightly more advanced. Suppose that you want to check that, for a given polynomial F, F(x+2)=F(x)+F(x+1) with the integer range {0,1…89} (so if you also check F(0)=F(1)=1, then F(100) would be the 100th Fibonacci number)
As polynomials, F(x+2)-F(x+1)-F(x) would not be exactly zero, as it could give arbitrary answers outside the range x={0,1…98}. But we can do something clever. In general, there is a rule that if a polynomial P is zero across some set S=\{x_1,x_2…x_n\} then it can be expressed as P(x)=Z(x)*H(x), where Z(x)=(x-x_1)*(x-x_2)*…*(x-x_n) and H(x) is also a polynomial. In other words, any polynomial that equals zero across some set is a (polynomial) multiple of the simplest (lowest-degree) polynomial that equals zero across that same set.
Why is this the case? It is a nice corollary of polynomial long division: the factor theorem. We know that, when dividing P(x) by Z(x), we will get a quotient Q(x) and a remainder R(x) is strictly less than that of Z(x). Since we know that P is zero on all of S, it means that R has to be zero on all of S as well. So we can simply compute R(x) via polynomial interpolation, since it's a polynomial of degree at most n-1 and we know n values (the zeros at S). Interpolating a polynomial with all zeroes gives the zero polynomial, thus R(x)=0 and H(x)=Q(x).
Going back to our example, if we have a polynomial F that encodes Fibonacci numbers (so F(x+2)=F(x)+F(x+1) across x=\{0,1…98\}), then I can convince you that F actually satisfies this condition by proving that the polynomial P(x)=F(x+2)-F(x+1)-F(x) is zero over that range, by giving you the quotient:
H(x)=\frac{F(x+2)-F(x+1)-F(x)}{Z(x)}
Where Z(x) = (x-0)*(x-1)*…*(x-98).
You can calculate Z(x) yourself (ideally you would have it precomputed), check the equation, and if the check passes then F(x) satisfies the condition!
Now, step back and notice what we did here. We converted a 100-step-long computation into a single equation with polynomials. Of course, proving the N'th Fibonacci number is not an especially useful task, especially since Fibonacci numbers have a closed form. But you can use exactly the same basic technique, just with some extra polynomials and some more complicated equations, to encode arbitrary computations with an arbitrarily large number of steps.
see part 3