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Nick Nolan

Nick Nolan

3 years ago

In five years, starting a business won't be hip.

More on Entrepreneurship/Creators

ANTHONY P.

ANTHONY P.

3 years ago

Startups are difficult. Streamlining the procedure for creating the following unicorn.

New ventures are exciting. It's fun to imagine yourself rich, successful, and famous (if that's your thing). How you'll help others and make your family proud. This excitement can pull you forward for years, even when you intuitively realize that the path you're on may not lead to your desired success.

Know when to change course. Switching course can mean pivoting or changing direction.

In this not-so-short blog, I'll describe the journey of building your dream. And how the journey might look when you think you're building your dream, but fall short of that vision. Both can feel similar in the beginning, but there are subtle differences.

Let’s dive in.

How an exciting journey to a dead end looks and feels.

You want to help many people. You're business-minded, creative, and ambitious. You jump into entrepreneurship. You're excited, free, and in control.

I'll use tech as an example because that's what I know best, but this applies to any entrepreneurial endeavor.

So you start learning the basics of your field, say coding/software development. You read books, take courses, and may even join a bootcamp. You start practicing, and the journey begins. Once you reach a certain level of skill (which can take months, usually 12-24), you gain the confidence to speak with others in the field and find common ground. You might attract a co-founder this way with time. You and this person embark on a journey (Tip: the idea you start with is rarely the idea you end with).

Amateur mistake #1: You spend months building a product before speaking to customers.

Building something pulls you forward blindly. You make mistakes, avoid customers, and build with your co-founder or small team in the dark for months, usually 6-12 months.

You're excited when the product launches. We'll be billionaires! The market won't believe it. This excites you and the team. Launch.

….

Nothing happens.

Some people may sign up out of pity, only to never use the product or service again.

You and the team are confused, discouraged and in denial. They don't get what we've built yet. We need to market it better, we need to talk to more investors, someone will understand our vision.

This is a hopeless path, and your denial could last another 6 months. If you're lucky, while talking to consumers and investors (which you should have done from the start), someone who has been there before would pity you and give you an idea to pivot into that can create income.

Suppose you get this idea and pivot your business. Again, you've just pivoted into something limited by what you've already built. It may be a revenue-generating idea, but it's rarely new. Now you're playing catch-up, doing something others are doing but you can do better. (Tip #2: Don't be late.) Your chances of winning are slim, and you'll likely never catch up.

You're finally seeing revenue and feel successful. You can compete, but if you're not a first mover, you won't earn enough over time. You'll get by or work harder than ever to earn what a skilled trade could provide. You didn't go into business to stress out and make $100,000 or $200,000 a year. When you can make the same amount by becoming a great software developer, electrician, etc.

You become stuck. Either your firm continues this way for years until you realize there isn't enough growth to recruit a strong team and remove yourself from day-to-day operations due to competition. Or a catastrophic economic event forces you to admit that what you were building wasn't new and unique and wouldn't get you where you wanted to be.

This realization could take 6-10 years. No kidding.

The good news is, you’ve learned a lot along the way and this information can be used towards your next venture (if you have the energy).

Key Lesson: Don’t build something if you aren’t one of the first in the space building it just for the sake of building something.

-

Let's discuss what it's like to build something that can make your dream come true.

Case 2: Building something the market loves is difficult but rewarding.

It starts with a problem that hasn't been adequately solved for a long time but is now solvable due to technology. Or a new problem due to a change in how things are done.

Let's examine each example.

Example #1: Mass communication. The problem is now solvable due to some technological breakthrough.

Twitter — One of the first web 2 companies that became successful with the rise of smart mobile computing.

People can share their real-time activities via mobile device with friends, family, and strangers. Web 2 and smartphones made it easy and fun.

Example #2: A new problem has emerged due to some change in the way things are conducted.

Zoom- A web-conferencing company that reached massive success due to the movement towards “work from home”, remote/hybrid work forces.

Online web conferencing allows for face-to-face communication.

-

These two examples show how to build a unicorn-type company. It's a mix of solving the right problem at the right time, either through a technological breakthrough that opens up new opportunities or by fundamentally changing how people do things.

Let's find these opportunities.

Start by examining problems, such as how the world has changed and how we can help it adapt. It can also be both. Start team brainstorming. Research technologies, current world-trends, use common sense, and make a list. Then, choose the top 3 that you're most excited about and seem most workable based on your skillsets, values, and passion.

Once you have this list, create the simplest MVP you can and test it with customers. The prototype can be as simple as a picture or diagram of user flow and end-user value. No coding required. Market-test. Twitter's version 1 was simple. It was a web form that asked, "What are you doing?" Then publish it from your phone. A global status update, wherever you are. Currently, this company has a $50 billion market cap.

Here's their MVP screenshot.

Small things grow. Tiny. Simplify.

Remember Frequency and Value when brainstorming. Your product is high frequency (Twitter, Instagram, Snapchat, TikTok) or high value (Airbnb for renting travel accommodations), or both (Gmail).

Once you've identified product ideas that meet the above criteria, they're simple, have a high frequency of use, or provide deep value. You then bring it to market in the simplest, most cost-effective way. You can sell a half-working prototype with imagination and sales skills. You need just enough of a prototype to convey your vision to a user or customer.

With this, you can approach real people. This will do one of three things: give you a green light to continue on your vision as is, show you that there is no opportunity and people won't use it, or point you in a direction that is a blend of what you've come up with and what the customer / user really wants, and you update the prototype and go back to the maze. Repeat until you have enough yeses and conviction to build an MVP.

Maddie Wang

Maddie Wang

3 years ago

Easiest and fastest way to test your startup idea!

Here's the fastest way to validate company concepts.

I squandered a year after dropping out of Stanford designing a product nobody wanted.

But today, I’m at 100k!

Differences:

I was designing a consumer product when I dropped out.

I coded MVP, got 1k users, and got YC interview.

Nice, huh?

WRONG!

Still coding and getting users 12 months later

WOULD PEOPLE PAY FOR IT? was the riskiest assumption I hadn't tested.

When asked why I didn't verify payment, I said,

Not-ready products. Now, nobody cares. The website needs work. Include this. Increase usage…

I feared people would say no.

After 1 year of pushing it off, my team told me they were really worried about the Business Model. Then I asked my audience if they'd buy my product.

So?

No, overwhelmingly.

I felt like I wasted a year building a product no one would buy.

Founders Cafe was the opposite.

Before building anything, I requested payment.

40 founders were interviewed.

Then we emailed Stanford, YC, and other top founders, asking them to join our community.

BOOM! 10/12 paid!

Without building anything, in 1 day I validated my startup's riskiest assumption. NOT 1 year.

Asking people to pay is one of the scariest things.

I understand.

I asked Stanford queer women to pay before joining my gay sorority.

I was afraid I'd turn them off or no one would pay.

Gay women, like those founders, were in such excruciating pain that they were willing to pay me upfront to help.

You can ask for payment (before you build) to see if people have the burning pain. Then they'll pay!

Examples from Founders Cafe members:

😮 Using a fake landing page, a college dropout tested a product. Paying! He built it and made $3m!

😮 YC solo founder faked a Powerpoint demo. 5 Enterprise paid LOIs. $1.5m raised, built, and in YC!

😮 A Harvard founder can convert Figma to React. 1 day, 10 customers. Built a tool to automate Figma -> React after manually fulfilling requests. 1m+

Bad example:

😭 Stanford Dropout Spends 1 Year Building Product Without Payment Validation

Some people build for a year and then get paying customers.

What I'm sharing is my experience and what Founders Cafe members have told me about validating startup ideas.

Don't waste a year like I did.

After my first startup failed, I planned to re-enroll at Stanford/work at Facebook.

After people paid, I quit for good.

I've hit $100k!

Hope this inspires you to request upfront payment! It'll change your life

Pat Vieljeux

Pat Vieljeux

3 years ago

Your entrepreneurial experience can either be a beautiful adventure or a living hell with just one decision.

Choose.

Bakhrom Tursunov — Unsplash

DNA makes us distinct.

We act alike. Most people follow the same road, ignoring differences. We remain quiet about our uniqueness for fear of exclusion (family, social background, religion). We live a more or less imposed life.

Off the beaten path, we stand out from the others. We obey without realizing we're sewing a shroud. We're told to do as everyone else and spend 40 years dreaming of a golden retirement and regretting not living.

“One of the greatest regrets in life is being what others would want you to be, rather than being yourself.” - Shannon L. Alder

Others dare. Again, few are creative; most follow the example of those who establish a business for the sake of entrepreneurship. To live.

They pick a potential market and model their MVP on an existing solution. Most mimic others, alter a few things, appear to be original, and end up with bland products, adding to an already crowded market.

SaaS, PaaS, etc. followed suit. It's reduced pricing, profitability, and product lifespan.

As competitors become more aggressive, their profitability diminishes, making life horrible for them and their employees. They fail to innovate, cut costs, and close their company.

Few of them look happy and fulfilled.

How did they do it?

The answer is unsettlingly simple.

They are themselves.

  • They start their company, propelled at first by a passion or maybe a calling.

  • Then, at their own pace, they create it with the intention of resolving a dilemma.

  • They assess what others are doing and consider how they might improve it.

  • In contrast to them, they respond to it in their own way by adding a unique personal touch. Therefore, it is obvious.

Originals, like their DNA, can't be copied. Or if they are, they're poorly printed. Originals are unmatched. Artist-like. True collectors only buy Picasso paintings by the master, not forgeries, no matter how good.

Imaginative people are constantly ahead. Copycats fall behind unless they innovate. They watch their competition continuously. Their solution or product isn't sexy. They hope to cash in on their copied product by flooding the market.

They're mostly pirates. They're short-sighted, unlike creators.

Creators see further ahead and have no rivals. They use copiers to confirm a necessity. To maintain their individuality, creators avoid copying others. They find copying boring. It's boring. They oppose plagiarism.

It's thrilling and inspiring.

It will also make them more able to withstand their opponents' tension. Not to mention roadblocks. For creators, impediments are games.

Others fear it. They race against the clock and fear threats that could interrupt their momentum since they lack inventiveness and their product has a short life cycle.

Creators have time on their side. They're dedicated. Clearly. Passionate booksellers will have their own bookstore. Their passion shows in their book choices. Only the ones they love.

The copier wants to display as many as possible, including mediocre authors, and will cut costs. All this to dominate the market. They're digging their own grave.

The bookseller is just one example. I could give you tons of them.

Closing remarks

Entrepreneurs might follow others or be themselves. They risk exhaustion trying to predict what their followers will do.

It's true.

Life offers choices.

Being oneself or doing as others do, with the possibility of regretting not expressing our uniqueness and not having lived.

“Be yourself; everyone else is already taken”. Oscar Wilde

The choice is yours.

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Sea Launch

Sea Launch

3 years ago

A guide to NFT pre-sales and whitelists

Before we dig through NFT whitelists and pre-sales, if you know absolutely nothing about NFTs, check our NFT Glossary.

What are pre-sales and whitelists on NFTs?

An NFT pre-sale, as the name implies, allows community members or early supporters of an NFT project to mint before the public, usually via a whitelist or mint pass.

Coin collectors can use mint passes to claim NFTs during the public sale. Because the mint pass is executed by “burning” an NFT into a specific crypto wallet, the collector is not concerned about gas price spikes.

A whitelist is used to approve a crypto wallet address for an NFT pre-sale. In a similar way to an early access list, it guarantees a certain number of crypto wallets can mint one (or more) NFT.

New NFT projects can do a pre-sale without a whitelist, but whitelists are good practice to avoid gas wars and a fair shot at minting an NFT before launching in competitive NFT marketplaces like Opensea, Magic Eden, or CNFT.

Should NFT projects do pre-sales or whitelists? 👇

The reasons to do pre-sales or a whitelist for NFT creators:

Time the market and gain traction.

Pre-sale or whitelists can help NFT projects gauge interest early on.

Whitelist spots filling up quickly is usually a sign of a successful launch, though it does not guarantee NFT longevity (more on that later). Also, full whitelists create FOMO and momentum for the public sale among non-whitelisted NFT collectors.

If whitelist signups are low or slow, projects may need to work on their vision, community, or product. Or the market is in a bear cycle. In either case, it aids NFT projects in market timing.

Reward the early NFT Community members.

Pre-sale and whitelists can help NFT creators reward early supporters.

First, by splitting the minting process into two phases, early adopters get a chance to mint one or more NFTs from their collection at a discounted or even free price.

Did you know that BAYC started at 0.08 eth each? A serum that allowed you to mint a Mutant Ape has become as valuable as the original BAYC.

(2) Whitelists encourage early supporters to help build a project's community in exchange for a slot or status. If you invite 10 people to the NFT Discord community, you get a better ranking or even a whitelist spot.

Pre-sale and whitelisting have become popular ways for new projects to grow their communities and secure future buyers.

Prevent gas wars.

Most new NFTs are created on the Ethereum blockchain, which has the highest transaction fees (also known as gas) (Solana, Cardano, Polygon, Binance Smart Chain, etc).

An NFT public sale is a gas war when a large number of NFT collectors (or bots) try to mint an NFT at the same time.

Competing collectors are willing to pay higher gas fees to prioritize their transaction and out-price others when upcoming NFT projects are hyped and very popular.

Pre-sales and whitelisting prevent gas wars by breaking the minting process into smaller batches of members or season launches.

The reasons to do pre-sales or a whitelists for NFT collectors:

How do I get on an NFT whitelist?

  1. Popular NFT collections act as a launchpad for other new or hyped NFT collections.

Example: Interfaces NFTs gives out 100 whitelist spots to Deadfellaz NFTs holders. Both NFT projects win. Interfaces benefit from Deadfellaz's success and brand equity.

In this case, to get whitelisted NFT collectors need to hold that specific NFT that is acting like a launchpad.

  1. A NFT studio or collection that launches a new NFT project and rewards previous NFT holders with whitelist spots or pre-sale access.

The whitelist requires previous NFT holders or community members.

NFT Alpha Groups are closed, small, tight-knit Discord servers where members share whitelist spots or giveaways from upcoming NFTs.

The benefit of being in an alpha group is getting information about new NFTs first and getting in on pre-sale/whitelist before everyone else.

There are some entry barriers to alpha groups, but if you're active in the NFT community, you'll eventually bump into, be invited to, or form one.

  1. A whitelist spot is awarded to members of an NFT community who are the most active and engaged.

This participation reward is the most democratic. To get a chance, collectors must work hard and play to their strengths.

Whitelisting participation examples:

  • Raffle, games and contest: NFT Community raffles, games, and contests. To get a whitelist spot, invite 10 people to X NFT Discord community.
  • Fan art: To reward those who add value and grow the community by whitelisting the best fan art and/or artists is only natural.
  • Giveaways: Lucky number crypto wallet giveaways promoted by an NFT community. To grow their communities and for lucky collectors, NFT projects often offer free NFT.
  • Activate your voice in the NFT Discord Community. Use voice channels to get NFT teams' attention and possibly get whitelisted.

The advantage of whitelists or NFT pre-sales.

Chainalysis's NFT stats quote is the best answer:

“Whitelisting isn’t just some nominal reward — it translates to dramatically better investing results. OpenSea data shows that users who make the whitelist and later sell their newly-minted NFT gain a profit 75.7% of the time, versus just 20.8% for users who do so without being whitelisted. Not only that, but the data suggests it’s nearly impossible to achieve outsized returns on minting purchases without being whitelisted.” Full report here.

Sure, it's not all about cash. However, any NFT collector should feel secure in their investment by owning a piece of a valuable and thriving NFT project. These stats help collectors understand that getting in early on an NFT project (via whitelist or pre-sale) will yield a better and larger return.

The downsides of pre-sales & whitelists for NFT creators.

Pre-sales and whitelist can cause issues for NFT creators and collectors.

NFT flippers

NFT collectors who only want to profit from early minting (pre-sale) or low mint cost (via whitelist). To sell the NFT in a secondary market like Opensea or Solanart, flippers go after the discounted price.

For example, a 1000 Solana NFT collection allows 100 people to mint 1 Solana NFT at 0.25 SOL. The public sale price for the remaining 900 NFTs is 1 SOL. If an NFT collector sells their discounted NFT for 0.5 SOL, the secondary market floor price is below the public mint.

This may deter potential NFT collectors. Furthermore, without a cap in the pre-sale minting phase, flippers can get as many NFTs as possible to sell for a profit, dumping them in secondary markets and driving down the floor price.

Hijacking NFT sites, communities, and pre-sales phase

People try to scam the NFT team and their community by creating oddly similar but fake websites, whitelist links, or NFT's Discord channel.

Established and new NFT projects must be vigilant to always make sure their communities know which are the official links, how a whitelist or pre-sale rules and how the team will contact (or not) community members.

Another way to avoid the scams around the pre-sale phase, NFT projects opt to create a separate mint contract for the whitelisted crypto wallets and then another for the public sale phase.

Scam NFT projects

We've seen a lot of mid-mint or post-launch rug pulls, indicating that some bad NFT projects are trying to scam NFT communities and marketplaces for quick profit. What happened to Magic Eden's launchpad recently will help you understand the scam.

We discussed the benefits and drawbacks of NFT pre-sales and whitelists for both projects and collectors. 

Finally, some practical tools and tips for finding new NFTs 👇

Tools & resources to find new NFT on pre-sale or to get on a whitelist:

In order to never miss an update, important pre-sale dates, or a giveaway, create a Tweetdeck or Tweeten Twitter dashboard with hyped NFT project pages, hashtags ( #NFTGiveaways , #NFTCommunity), or big NFT influencers.

Search for upcoming NFT launches that have been vetted by the marketplace and try to get whitelisted before the public launch.

Save-timing discovery platforms like sealaunch.xyz for NFT pre-sales and upcoming launches. How can we help 100x NFT collectors get projects? A project's official social media links, description, pre-sale or public sale dates, price and supply. We're also working with Dune on NFT data analysis to help NFT collectors make better decisions.

Don't invest what you can't afford to lose because a) the project may fail or become rugged. Find NFTs projects that you want to be a part of and support.

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Jon Brosio

Jon Brosio

3 years ago

You can learn more about marketing from these 8 copywriting frameworks than from a college education.

Email, landing pages, and digital content

Photo by Ron Lach from Pexels

Today's most significant skill:

Copywriting.

Unfortunately, most people don't know how to write successful copy because they weren't taught in school.

I've been obsessed with copywriting for two years. I've read 15 books, completed 3 courses, and studied internet's best digital entrepreneurs.

Here are 8 copywriting frameworks that educate more than a four-year degree.

1. Feature — Advantage — Benefit (F.A.B)

This is the most basic copywriting foundation. Email marketing, landing page copy, and digital video ads can use it.

F.A.B says:

  • How it works (feature)

  • which is helpful (advantage)

  • What's at stake (benefit)

The Hustle uses this framework on their landing page to convince people to sign up:

Courtesy | Thehustle.co

2. P. A. S. T. O. R.

This framework is for longer-form copywriting. PASTOR uses stories to engage with prospects. It explains why people should buy this offer.

PASTOR means:

  • Problem

  • Amplify

  • Story

  • Testimonial

  • Offer

  • Response

Dan Koe's landing page is a great example. It shows PASTOR frame-by-frame.

Courtesy | Dan Koe

3. Before — After — Bridge

Before-after-bridge is a copywriting framework that draws attention and shows value quickly.

This framework highlights:

  • where you are

  • where you want to be

  • how to get there

Works great for: Email threads/landing pages

Zain Kahn utilizes this framework to write viral threads.

Courtesy | Zain Kahn

4. Q.U.E.S.T

QUEST is about empathetic writing. You know their issues, obstacles, and headaches. This allows coverups.

QUEST:

  • Qualifies

  • Understands

  • Educates

  • Stimulates

  • Transitions

Tom Hirst's landing page uses the QUEST framework.

Courtesy | Tom Hirst

5. The 4P’s model

The 4P’s approach pushes your prospect to action. It educates and persuades quickly.

4Ps:

  • The problem the visitor is dealing with

  • The promise that will help them

  • The proof the promise works

  • push towards action

Mark Manson is a bestselling author, digital creator, and pop-philosopher. He's also a great copywriter, and his membership offer uses the 4P’s framework.

Courtesy | Mark Manson

6. Problem — Agitate — Solution (P.A.S)

Up-and-coming marketers should understand problem-agitate-solution copywriting. Once you understand one structure, others are easier. It drives passion and presents a clear solution.

PAS outlines:

  • The issue the visitor is having

  • It then intensifies this issue through emotion.

  • finally offers an answer to that issue (the offer)

The customer's story loops. Nicolas Cole and Dickie Bush use PAS to promote Ship 30 for 30.

Courtesy | ship30for30.com

7. Star — Story — Solution (S.S.S)

PASTOR + PAS = star-solution-story. Like PAS, it employs stories to persuade.

S.S.S. is effective storytelling:

  • Star: (Person had a problem)

  • Story: (until they had a breakthrough)

  • Solution: (That created a transformation)

Ali Abdaal is a YouTuber with a great S.S.S copy.

Courtesy | Ali Abdaal

8. Attention — Interest — Desire — Action

AIDA is another classic. This copywriting framework is great for fast-paced environments (think all digital content on Linkedin, Twitter, Medium, etc.).

It works with:

  • Page landings

  • writing on thread

  • Email

It's a good structure since it's concise, attention-grabbing, and action-oriented.

Shane Martin, Twitter's creator, uses this approach to create viral content.

Courtesy | Shane Martin

TL;DR

8 copywriting frameworks that teach marketing better than a four-year degree

  • Feature-advantage-benefit

  • Before-after-bridge

  • Star-story-solution

  • P.A.S.T.O.R

  • Q.U.E.S.T

  • A.I.D.A

  • P.A.S

  • 4P’s

Vitalik

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