More on Society & Culture

DC Palter
2 years ago
Why Are There So Few Startups in Japan?
Japan's startup challenge: 7 reasons
Every day, another Silicon Valley business is bought for a billion dollars, making its founders rich while growing the economy and improving consumers' lives.
Google, Amazon, Twitter, and Medium dominate our daily lives. Tesla automobiles and Moderna Covid vaccinations.
The startup movement started in Silicon Valley, California, but the rest of the world is catching up. Global startup buzz is rising. Except Japan.
644 of CB Insights' 1170 unicorns—successful firms valued at over $1 billion—are US-based. China follows with 302 and India third with 108.
Japan? 6!
1% of US startups succeed. The third-largest economy is tied with small Switzerland for startup success.
Mexico (8), Indonesia (12), and Brazil (12) have more successful startups than Japan (16). South Korea has 16. Yikes! Problem?
Why Don't Startups Exist in Japan More?
Not about money. Japanese firms invest in startups. To invest in startups, big Japanese firms create Silicon Valley offices instead of Tokyo.
Startups aren't the issue either. Local governments are competing to be Japan's Shirikon Tani, providing entrepreneurs financing, office space, and founder visas.
Startup accelerators like Plug and Play in Tokyo, Osaka, and Kyoto, the Startup Hub in Kobe, and Google for Startups are many.
Most of the companies I've encountered in Japan are either local offices of foreign firms aiming to expand into the Japanese market or small businesses offering local services rather than disrupting a staid industry with new ideas.
There must be a reason Japan can develop world-beating giant corporations like Toyota, Nintendo, Shiseido, and Suntory but not inventive startups.
Culture, obviously. Japanese culture excels in teamwork, craftsmanship, and quality, but it hates moving fast, making mistakes, and breaking things.
If you have a brilliant idea in Silicon Valley, quit your job, get money from friends and family, and build a prototype. To fund the business, you approach angel investors and VCs.
Most non-startup folks don't aware that venture capitalists don't want good, profitable enterprises. That's wonderful if you're developing a solid small business to consult, open shops, or make a specialty product. However, you must pay for it or borrow money. Venture capitalists want moon rockets. Silicon Valley is big or bust. Almost 90% will explode and crash. The few successes are remarkable enough to make up for the failures.
Silicon Valley's high-risk, high-reward attitude contrasts with Japan's incrementalism. Japan makes the best automobiles and cleanrooms, but it fails to produce new items that grow the economy.
Changeable? Absolutely. But, what makes huge manufacturing enterprises successful and what makes Japan a safe and comfortable place to live are inextricably connected with the lack of startups.
Barriers to Startup Development in Japan
These are the 7 biggest obstacles to Japanese startup success.
Unresponsive Employment Market
While the lifelong employment system in Japan is evolving, the average employee stays at their firm for 12 years (15 years for men at large organizations) compared to 4.3 years in the US. Seniority, not experience or aptitude, determines career routes, making it tough to quit a job to join a startup and then return to corporate work if it fails.
Conservative Buyers
Even if your product is buggy and undocumented, US customers will migrate to a cheaper, superior one. Japanese corporations demand perfection from their trusted suppliers and keep with them forever. Startups need income fast, yet product evaluation takes forever.
Failure intolerance
Japanese business failures harm lives. Failed forever. It hinders risk-taking. Silicon Valley embraces failure. Build another startup if your first fails. Build a third if that fails. Every setback is viewed as a learning opportunity for success.
4. No Corporate Purchases
Silicon Valley industrial giants will buy fast-growing startups for a lot of money. Many huge firms have stopped developing new goods and instead buy startups after the product is validated.
Japanese companies prefer in-house product development over startup acquisitions. No acquisitions mean no startup investment and no investor reward.
Startup investments can also be monetized through stock market listings. Public stock listings in Japan are risky because the Nikkei was stagnant for 35 years while the S&P rose 14x.
5. Social Unity Above Wealth
In Silicon Valley, everyone wants to be rich. That creates a competitive environment where everyone wants to succeed, but it also promotes fraud and societal problems.
Japan values communal harmony above individual success. Wealthy folks and overachievers are avoided. In Japan, renegades are nearly impossible.
6. Rote Learning Education System
Japanese high school graduates outperform most Americans. Nonetheless, Japanese education is known for its rote memorization. The American system, which fails too many kids, emphasizes creativity to create new products.
Immigration.
Immigrants start 55% of successful Silicon Valley firms. Some come for university, some to escape poverty and war, and some are recruited by Silicon Valley startups and stay to start their own.
Japan is difficult for immigrants to start a business due to language barriers, visa restrictions, and social isolation.
How Japan Can Promote Innovation
Patchwork solutions to deep-rooted cultural issues will not work. If customers don't buy things, immigration visas won't aid startups. Startups must have a chance of being acquired for a huge sum to attract investors. If risky startups fail, employees won't join.
Will Japan never have a startup culture?
Once a consensus is reached, Japan changes rapidly. A dwindling population and standard of living may lead to such consensus.
Toyota and Sony were firms with renowned founders who used technology to transform the world. Repeatable.
Silicon Valley is flawed too. Many people struggle due to wealth disparities, job churn and layoffs, and the tremendous ups and downs of the economy caused by stock market fluctuations.
The founders of the 10% successful startups are heroes. The 90% that fail and return to good-paying jobs with benefits are never mentioned.
Silicon Valley startup culture and Japanese corporate culture are opposites. Each have pros and cons. Big Japanese corporations make the most reliable, dependable, high-quality products yet move too slowly. That's good for creating cars, not social networking apps.
Can innovation and success be encouraged without eroding social cohesion? That can motivate software firms to move fast and break things while recognizing the beauty and precision of expert craftsmen? A hybrid culture where Japan can make the world's best and most original items. Hopefully.

Scott Galloway
3 years ago
Attentive
From oil to attention.
Oil has been the most important commodity for a century. It's sparked wars. Pearl Harbor was a preemptive strike to guarantee Japanese access to Indonesian oil, and it made desert tribes rich. Oil's heyday is over. From oil to attention.
We talked about an information economy. In an age of abundant information, what's scarce? Attention. Scale of the world's largest enterprises, wealth of its richest people, and power of governments all stem from attention extraction, monetization, and custody.
Attention-grabbing isn't new. Humans have competed for attention and turned content into wealth since Aeschylus' Oresteia. The internal combustion engine, industrial revolutions in mechanization and plastics, and the emergence of a mobile Western lifestyle boosted oil. Digitization has put wells in pockets, on automobile dashboards, and on kitchen counters, drilling for attention.
The most valuable firms are attention-seeking enterprises, not oil companies. Big Tech dominates the top 4. Tech and media firms are the sheikhs and wildcatters who capture our attention. Blood will flow as the oil economy rises.
Attention to Detail
More than IT and media companies compete for attention. Podcasting is a high-growth, low-barrier-to-entry chance for newbies to gain attention and (for around 1%) make money. Conferences are good for capturing in-person attention. Salesforce paid $30 billion for Slack's dominance of workplace attention, while Spotify is transforming music listening attention into a media platform.
Conferences, newsletters, and even music streaming are artisan projects. Even 130,000-person Comic Con barely registers on the attention economy's Richter scale. Big players have hundreds of millions of monthly users.
Supermajors
Even titans can be disrupted in the attention economy. TikTok is fracking king Chesapeake Energy, a rule-breaking insurgent with revolutionary extraction technologies. Attention must be extracted, processed, and monetized. Innovators disrupt the attention economy value chain.
Attention pre-digital Entrepreneurs commercialized intriguing or amusing stuff like a newspaper or TV show through subscriptions and ads. Digital storage and distribution's limitless capacity drove the initial wave of innovation. Netflix became dominant by releasing old sitcoms and movies. More ad-free content gained attention. By 2016, Netflix was greater than cable TV. Linear scale, few network effects.
Social media introduced two breakthroughs. First, users produced and paid for content. Netflix's economics are dwarfed by TikTok and YouTube, where customers create the content drill rigs that the platforms monetize.
Next, social media businesses expanded content possibilities. Twitter, Facebook, and Reddit offer traditional content, but they transform user comments into more valuable (addictive) emotional content. By emotional resonance, I mean they satisfy a craving for acceptance or anger us. Attention and emotion are mined from comments/replies, piss-fights, and fast-brigaded craziness. Exxon has turned exhaust into heroin. Should we be so linked without a commensurate presence? You wouldn't say this in person. Anonymity allows fraudulent accounts and undesirable actors, which platforms accept to profit from more pollution.
FrackTok
A new entrepreneur emerged as ad-driven social media anger contaminated the water table. TikTok is remaking the attention economy. Short-form video platform relies on user-generated content, although delivery is narrower and less social.
Netflix grew on endless options. Choice requires cognitive effort. TikTok is the least demanding platform since TV. App video plays when opened. Every video can be skipped with a swipe. An algorithm watches how long you watch, what you finish, and whether you like or follow to create a unique streaming network. You can follow creators and respond, but the app is passive. TikTok's attention economy recombination makes it apex predator. The app has more users than Facebook and Instagram combined. Among teens, it's overtaking the passive king, TV.
Externalities
Now we understand fossil fuel externalities. A carbon-based economy has harmed the world. Fracking brought large riches and rebalanced the oil economy, but at a cost: flammable water, earthquakes, and chemical leaks.
TikTok has various concerns associated with algorithmically generated content and platforms. A Wall Street Journal analysis discovered new accounts listed as belonging to 13- to 15-year-olds would swerve into rabbitholes of sex- and drug-related films in mere days. TikTok has a unique externality: Chinese Communist Party ties. Our last two presidents realized the relationship's perils. Concerned about platform's propaganda potential.
No evidence suggests the CCP manipulated information to harm American interests. A headjack implanted on America's youth, who spend more time on TikTok than any other network, connects them to a neural network that may be modified by the CCP. If the product and ownership can't be separated, the app should be banned. Putting restrictions near media increases problems. We should have a reciprocal approach with China regarding media firms. Ban TikTok
It was a conference theme. I anticipated Axel Springer CEO Mathias Döpfner to say, "We're watching them." (That's CEO protocol.) TikTok should be outlawed in every democracy as an espionage tool. Rumored regulations could lead to a ban, and FCC Commissioner Brendan Carr pushes for app store prohibitions. Why not restrict Chinese propaganda? Some disagree: Several renowned tech writers argued my TikTok diatribe last week distracted us from privacy and data reform. The situation isn't zero-sum. I've warned about Facebook and other tech platforms for years. Chewing gum while walking is possible.
The Future
Is TikTok the attention-economy titans' final evolution? The attention economy acts like it. No original content. CNN+ was unplugged, Netflix is losing members and has lost 70% of its market cap, and households are canceling cable and streaming subscriptions in historic numbers. Snap Originals closed in August after YouTube Originals in January.
Everyone is outTik-ing the Tok. Netflix debuted Fast Laughs, Instagram Reels, YouTube Shorts, Snap Spotlight, Roku The Buzz, Pinterest Watch, and Twitter is developing a TikTok-like product. I think they should call it Vine. Just a thought.
Meta's internal documents show that users spend less time on Instagram Reels than TikTok. Reels engagement is dropping, possibly because a third of the videos were generated elsewhere (usually TikTok, complete with watermark). Meta has tried to downrank these videos, but they persist. Users reject product modifications. Kim Kardashian and Kylie Jenner posted a meme urging Meta to Make Instagram Instagram Again, resulting in 312,000 signatures. Mark won't hear the petition. Meta is the fastest follower in social (see Oculus and legless hellscape fever nightmares). Meta's stock is at a five-year low, giving those who opposed my demands to break it up a compelling argument.
Blue Pill
TikTok's short-term dominance in attention extraction won't be stopped by anyone who doesn't hear Hail to the Chief every time they come in. Will TikTok still be a supermajor in five years? If not, YouTube will likely rule and protect Kings Landing.
56% of Americans regularly watch YouTube. Compared to Facebook and TikTok, 95% of teens use Instagram. YouTube users upload more than 500 hours of video per minute, a number that's likely higher today. Last year, the platform garnered $29 billion in advertising income, equivalent to Netflix's total.
Business and biology both value diversity. Oil can be found in the desert, under the sea, or in the Arctic. Each area requires a specific ability. Refiners turn crude into gas, lubricants, and aspirin. YouTube's variety is unmatched. One-second videos to 12-hour movies. Others are studio-produced. (My Bill Maher appearance was edited for YouTube.)
You can dispute in the comment section or just stream videos. YouTube is used for home improvement, makeup advice, music videos, product reviews, etc. You can load endless videos on a topic or creator, subscribe to your favorites, or let the suggestion algo take over. YouTube relies on user content, but it doesn't wait passively. Strategic partners advise 12,000 creators. According to a senior director, if a YouTube star doesn’t post once week, their manager is “likely to know why.”
YouTube's kevlar is its middle, especially for creators. Like TikTok, users can start with low-production vlogs and selfie videos. As your following expands, so does the scope of your production, bringing longer videos, broadcast-quality camera teams and performers, and increasing prices. MrBeast, a YouTuber, is an example. MrBeast made gaming videos and YouTube drama comments.
Donaldson's YouTube subscriber base rose. MrBeast invests earnings to develop impressive productions. His most popular video was a $3.5 million Squid Game reenactment (the cost of an episode of Mad Men). 300 million people watched. TikTok's attention-grabbing tech is too limiting for this type of material. Now, Donaldson is focusing on offline energy with a burger restaurant and cloud kitchen enterprise.
Steps to Take
Rapid wealth growth has externalities. There is no free lunch. OK, maybe caffeine. The externalities are opaque, and the parties best suited to handle them early are incentivized to construct weapons of mass distraction to postpone and obfuscate while achieving economic security for themselves and their families. The longer an externality runs unchecked, the more damage it causes and the more it costs to fix. Vanessa Pappas, TikTok's COO, didn't shine before congressional hearings. Her comms team over-consulted her and said ByteDance had no headquarters because it's scattered. Being full of garbage simply promotes further anger against the company and the awkward bond it's built between the CCP and a rising generation of American citizens.
This shouldn't distract us from the (still existent) harm American platforms pose to our privacy, teenagers' mental health, and civic dialogue. Leaders of American media outlets don't suffer from immorality but amorality, indifference, and dissonance. Money rain blurs eyesight.
Autocratic governments that undermine America's standing and way of life are immoral. The CCP has and will continue to use all its assets to harm U.S. interests domestically and abroad. TikTok should be spun to Western investors or treated the way China treats American platforms: kicked out.
So rich,

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.
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Jenn Leach
3 years ago
This clever Instagram marketing technique increased my sales to $30,000 per month.
No Paid Ads Required
I had an online store. After a year of running the company alongside my 9-to-5, I made enough to resign.
That day was amazing.
This Instagram marketing plan helped the store succeed.
How did I increase my sales to five figures a month without using any paid advertising?
I used customer event marketing.
I'm not sure this term exists. I invented it to describe what I was doing.
Instagram word-of-mouth, fan engagement, and interaction drove sales.
If a customer liked or disliked a product, the buzz would drive attention to the store.
I used customer-based events to increase engagement and store sales.
Success!
Here are the weekly Instagram customer events I coordinated while running my business:
Be the Buyer Days
Flash sales
Mystery boxes
Be the Buyer Days: How do they work?
Be the Buyer Days are exactly that.
You choose a day to share stock selections with social media followers.
This is an easy approach to engaging customers and getting fans enthusiastic about new releases.
First, pick a handful of items you’re considering ordering. I’d usually pick around 3 for Be the Buyer Day.
Then I'd poll the crowd on Instagram to vote on their favorites.
This was before Instagram stories, polls, and all the other cool features Instagram offers today. I think using these tools now would make this event even better.
I'd ask customers their favorite back then.
The growing comments excited customers.
Then I'd declare the winner, acquire the products, and start selling it.
How do flash sales work?
I mostly ran flash sales.
You choose a limited number of itemsdd for a few-hour sale.
We wanted most sales to result in sold-out items.
When an item sells out, it contributes to the sensation of scarcity and can inspire customers to visit your store to buy a comparable product, join your email list, become a fan, etc.
We hoped they'd act quickly.
I'd hold flash deals twice a week, which generated scarcity and boosted sales.
The store had a few thousand Instagram followers when I started flash deals.
Each flash sale item would make $400 to $600.
$400 x 3= $1,200
That's $1,200 on social media!
Twice a week, you'll make roughly $10K a month from Instagram.
$1,200/day x 8 events/month=$9,600
Flash sales did great.
We held weekly flash deals and sent social media and email reminders. That’s about it!
How are mystery boxes put together?
All you do is package a box of store products and sell it as a mystery box on TikTok or retail websites.
A $100 mystery box would cost $30.
You're discounting high-value boxes.
This is a clever approach to get rid of excess inventory and makes customers happy.
It worked!
Be the Buyer Days, flash deals, and mystery boxes helped build my company without paid advertisements.
All companies can use customer event marketing. Involving customers and providing an engaging environment can boost sales.
Try it!

Jari Roomer
3 years ago
Successful people have this one skill.
Without self-control, you'll waste time chasing dopamine fixes.
I found a powerful quote in Tony Robbins' Awaken The Giant Within:
“Most of the challenges that we have in our personal lives come from a short-term focus” — Tony Robbins
Most people are short-term oriented, but highly successful people are long-term oriented.
Successful people act in line with their long-term goals and values, while the rest are distracted by short-term pleasures and dopamine fixes.
Instant gratification wrecks lives
Instant pleasure is fleeting. Quickly fading effects leave you craving more stimulation.
Before you know it, you're in a cycle of quick fixes. This explains binging on food, social media, and Netflix.
These things cause a dopamine spike, which is entertaining. This dopamine spike crashes quickly, leaving you craving more stimulation.
It's fine to watch TV or play video games occasionally. Problems arise when brain impulses aren't controlled. You waste hours chasing dopamine fixes.
Instant gratification becomes problematic when it interferes with long-term goals, happiness, and life fulfillment.
Most rewarding things require delay
Life's greatest rewards require patience and delayed gratification. They must be earned through patience, consistency, and effort.
Ex:
A fit, healthy body
A deep connection with your spouse
A thriving career/business
A healthy financial situation
These are some of life's most rewarding things, but they take work and patience. They all require the ability to delay gratification.
To have a healthy bank account, you must save (and invest) a large portion of your monthly income. This means no new tech or clothes.
If you want a fit, healthy body, you must eat better and exercise three times a week. So no fast food and Netflix.
It's a battle between what you want now and what you want most.
Successful people choose what they want most over what they want now. It's a major difference.
Instant vs. delayed gratification
Most people subconsciously prefer instant rewards over future rewards, even if the future rewards are more significant.
We humans aren't logical. Emotions and instincts drive us. So we act against our goals and values.
Fortunately, instant gratification bias can be overridden. This is a modern superpower. Effective methods include:
#1: Train your brain to handle overstimulation
Training your brain to function without constant stimulation is a powerful change. Boredom can lead to long-term rewards.
Unlike impulsive shopping, saving money is boring. Having lots of cash is amazing.
Compared to video games, deep work is boring. A successful online business is rewarding.
Reading books is boring compared to scrolling through funny videos on social media. Knowledge is invaluable.
You can't do these things if your brain is overstimulated. Your impulses will control you. To reduce overstimulation addiction, try:
Daily meditation (10 minutes is enough)
Daily study/work for 90 minutes (no distractions allowed)
First hour of the day without phone, social media, and Netflix
Nature walks, journaling, reading, sports, etc.
#2: Make Important Activities Less Intimidating
Instant gratification helps us cope with stress. Starting a book or business can be intimidating. Video games and social media offer a quick escape in such situations.
Make intimidating tasks less so. Break them down into small tasks. Start a new business/side-hustle by:
Get domain name
Design website
Write out a business plan
Research competition/peers
Approach first potential client
Instead of one big mountain, divide it into smaller sub-tasks. This makes a task easier and less intimidating.
#3: Plan ahead for important activities
Distractions will invade unplanned time. Your time is dictated by your impulses, which are usually Netflix, social media, fast food, and video games. It wants quick rewards and dopamine fixes.
Plan your days and be proactive with your time. Studies show that scheduling activities makes you 3x more likely to do them.
To achieve big goals, you must plan. Don't gamble.
Want to get fit? Schedule next week's workouts. Want a side-job? Schedule your work time.

Vitalik
3 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
