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Jenn Leach

Jenn Leach

3 years ago

I created a faceless TikTok account. Six months later.

More on Entrepreneurship/Creators

Bastian Hasslinger

Bastian Hasslinger

3 years ago

Before 2021, most startups had excessive valuations. It is currently causing issues.

Higher startup valuations are often favorable for all parties. High valuations show a business's potential. New customers and talent are attracted. They earn respect.

Everyone benefits if a company's valuation rises.

Founders and investors have always been incentivized to overestimate a company's value.

Post-money valuations were inflated by 2021 market expectations and the valuation model's mechanisms.

Founders must understand both levers to handle a normalizing market.

2021, the year of miracles

2021 must've seemed miraculous to entrepreneurs, employees, and VCs. Valuations rose, and funding resumed after the first Covid-19 epidemic caution.

In 2021, VC investments increased from $335B to $643B. 518 new worldwide unicorns vs. 134 in 2020; 951 US IPOs vs. 431.

Things can change quickly, as 2020-21 showed.

Rising interest rates, geopolitical developments, and normalizing technology conditions drive down share prices and tech company market caps in 2022. Zoom, the poster-child of early lockdown success, is down 37% since 1st Jan.

Once-inflated valuations can become a problem in a normalizing market, especially for founders, employees, and early investors.

the reason why startups are always overvalued

To see why inflated valuations are a problem, consider one of its causes.

Private company values only fluctuate following a new investment round, unlike publicly-traded corporations. The startup's new value is calculated simply:

(Latest round share price) x (total number of company shares)

This is the industry standard Post-Money Valuation model.

Let’s illustrate how it works with an example. If a VC invests $10M for 1M shares (at $10/share), and the company has 10M shares after the round, its Post-Money Valuation is $100M (10/share x 10M shares).

This approach might seem like the most natural way to assess a business, but the model often unintentionally overstates the underlying value of the company even if the share price paid by the investor is fair. All shares aren't equal.

New investors in a corporation will always try to minimize their downside risk, or the amount they lose if things go wrong. New investors will try to negotiate better terms and pay a premium.

How the value of a struggling SpaceX increased

SpaceX's 2008 Series D is an example. Despite the financial crisis and unsuccessful rocket launches, the company's Post-Money Valuation was 36% higher after the investment round. Why?

Series D SpaceX shares were protected. In case of liquidation, Series D investors were guaranteed a 2x return before other shareholders.

Due to downside protection, investors were willing to pay a higher price for this new share class.

The Post-Money Valuation model overpriced SpaceX because it viewed all the shares as equal (they weren't).

Why entrepreneurs, workers, and early investors stand to lose the most

Post-Money Valuation is an effective and sufficient method for assessing a startup's valuation, despite not taking share class disparities into consideration.

In a robust market, where the firm valuation will certainly expand with the next fundraising round or exit, the inflated value is of little significance.

Fairness endures. If a corporation leaves at a greater valuation, each stakeholder will receive a proportional distribution. (i.e., 5% of a $100M corporation yields $5M).

SpaceX's inherent overvaluation was never a problem. Had it been sold for less than its Post-Money Valuation, some shareholders, including founders, staff, and early investors, would have seen their ownership drop.

The unforgiving world of 2022

In 2022, founders, employees, and investors who benefited from inflated values will face below-valuation exits and down-rounds.

For them, 2021 will be a curse, not a blessing.

Some tech giants are worried. Klarna's valuation fell from $45B (Oct 21) to $30B (Jun 22), Canvas from $40B to $27B, and GoPuffs from $17B to $8.3B.

Shazam and Blue Apron have to exit or IPO at a cheaper price. Premium share classes are protected, while others receive less. The same goes for bankrupts.

Those who continue at lower valuations will lose reputation and talent. When their value declines by half, generous employee stock options become less enticing, and their ability to return anything is questioned.

What can we infer about the present situation?

Such techniques to enhance your company's value or stop a normalizing market are fiction.

The current situation is a painful reminder for entrepreneurs and a crucial lesson for future firms.

The devastating market fall of the previous six months has taught us one thing:

  1. Keep in mind that any valuation is speculative. Money Post A startup's valuation is a highly simplified approximation of its true value, particularly in the early phases when it lacks significant income or a cutting-edge product. It is merely a projection of the future and a hypothetical meter. Until it is achieved by an exit, a valuation is nothing more than a number on paper.

  2. Assume the value of your company is lower than it was in the past. Your previous valuation might not be accurate now due to substantial changes in the startup financing markets. There is little reason to think that your company's value will remain the same given the 50%+ decline in many newly listed IT companies. Recognize how the market situation is changing and use caution.

  3. Recognize the importance of the stake you hold. Each share class has a unique value that varies. Know the sort of share class you own and how additional contractual provisions affect the market value of your security. Frameworks have been provided by Metrick and Yasuda (Yale & UC) and Gornall and Strebulaev (Stanford) for comprehending the terms that affect investors' cash-flow rights upon withdrawal. As a result, you will be able to more accurately evaluate your firm and determine the worth of each share class.

  4. Be wary of approving excessively protective share terms.
    The trade-offs should be considered while negotiating subsequent rounds. Accepting punitive contractual terms could first seem like a smart option in order to uphold your inflated worth, but you should proceed with caution. Such provisions ALWAYS result in misaligned shareholders, with common shareholders (such as you and your staff) at the bottom of the list.

Matthew O'Riordan

Matthew O'Riordan

3 years ago

Trends in SaaS Funding from 2016 to 2022

Christopher Janz of Point Nine Capital created the SaaS napkin in 2016. This post shows how founders have raised cash in the last 6 years. View raw data.

Round size

Unsurprisingly, round sizes have expanded and will taper down in 2022. In 2016, pre-seed rounds were $200k to $500k; currently, they're $1-$2m. Despite the macroeconomic scenario, Series A have expanded from $3m to $12m in 2016 to $6m and $18m in 2022.

Generated from raw data for Seed to Series B from 2016–2022

Valuation

There are hints that valuations are rebounding this year. Pre-seed valuations in 2022 are $12m from $3m in 2016, and Series B prices are $270m from $100m in 2016.

Generated from raw data for Seed to Series B from 2016–2022

Compared to public SaaS multiples, Series B valuations more closely reflect the market, but Seed and Series A prices seem to be inflated regardless of the market.

Source: CapitalIQ as of 13-May-2022

I'd like to know how each annual cohort performed for investors, based on the year they invested and the valuations. I can't access this information.

ARR

Seed firms' ARR forecasts have risen from $0 to $0.6m to $0 to $1m. 2016 expected $1.2m to $3m, 2021 $0.5m to $4m, and this year $0.5m to $2.5m, suggesting that Series A firms may raise with less ARR today. Series B minutes fell from $4.2m to $3m.

Generated from raw data for Seed to Series B from 2016–2022

Capitalization Rate

2022 is the year that VCs start discussing capital efficiency in portfolio meetings. Given the economic shift in the markets and the stealthy VC meltdown, it's not surprising. Christopher Janz added capital efficiency to the SaaS Napkin as a new statistic for Series A (3.5x) and Series B. (2.5x). Your investors must live under a rock if they haven't asked about capital efficiency. If you're unsure:

The Capital Efficiency Ratio is the ratio of how much a company has spent growing revenue and how much they’re receiving in return. It is the broadest measure of company effectiveness in generating ARR

What next?

No one knows what's next, including me. All startup and growing enterprises around me are tightening their belts and extending their runways in anticipation of a difficult fundraising ride. If you're wanting to raise money but can wait, wait till the market is more stable and access to money is easier.

Micah Daigle

Micah Daigle

3 years ago

Facebook is going away. Here are two explanations for why it hasn't been replaced yet.

And tips for anyone trying.

We see the same story every few years.

BREAKING NEWS: [Platform X] launched a social network. With Facebook's reputation down, the new startup bets millions will switch.

Despite the excitement surrounding each new platform (Diaspora, Ello, Path, MeWe, Minds, Vero, etc.), no major exodus occurred.

Snapchat and TikTok attracted teens with fresh experiences (ephemeral messaging and rapid-fire videos). These features aren't Facebook, even if Facebook replicated them.

Facebook's core is simple: you publish items (typically text/images) and your friends (generally people you know IRL) can discuss them.

It's cool. Sometimes I don't want to, but sh*t. I like it.

Because, well, I like many folks I've met. I enjoy keeping in touch with them and their banter.

I dislike Facebook's corporation. I've been cautiously optimistic whenever a Facebook-killer surfaced.

None succeeded.

Why? Two causes, I think:

People couldn't switch quickly enough, which is reason #1

Your buddies make a social network social.

Facebook started in self-contained communities (college campuses) then grew outward. But a new platform can't.

If we're expected to leave Facebook, we want to know that most of our friends will too.

Most Facebook-killers had bottlenecks. You have to waitlist or jump through hoops (e.g. setting up a server).

Same outcome. Upload. Chirp.

After a week or two of silence, individuals returned to Facebook.

Reason #2: The fundamental experience was different.

Even when many of our friends joined in the first few weeks, it wasn't the same.

There were missing features or a different UX.

Want to reply with a meme? No photos in comments yet. (Trying!)

Want to tag a friend? Nope, sorry. 2019!

Want your friends to see your post? You must post to all your friends' servers. Good luck!

It's difficult to introduce a platform with 100% of the same features as one that's been there for 20 years, yet customers want a core experience.

If you can't, they'll depart.

The causes that led to the causes

Having worked on software teams for 14+ years, I'm not surprised by these challenges. They are a natural development of a few tech sector meta-problems:

Lean startup methodology

Silicon Valley worships lean startup. It's a way of developing software that involves testing a stripped-down version with a limited number of people before selecting what to build.

Billion people use Facebook's functions. They aren't tested. It must work right away*

*This may seem weird to software people, but it's how non-software works! You can't sell a car without wheels.

2. Creativity

Startup entrepreneurs build new things, not copies. I understand. Reinventing the wheel is boring.

We know what works. Different experiences raise adoption friction. Once millions have transferred, more features (and a friendlier UX) can be implemented.

3. Cost scaling

True. Building a product that can sustain hundreds of millions of users in weeks is expensive and complex.

Your lifeboats must have the same capacity as the ship you're evacuating. It's required.

4. Pure ideologies

People who work on Facebook-alternatives are (understandably) critical of Facebook.

They build an open-source, fully-distributed, data-portable, interface-customizable, offline-capable, censorship-proof platform.

Prioritizing these aims can prevent replicating the straightforward experience users expect. Github, not Facebook, is for techies only.

What about the business plan, though?

Facebook-killer attempts have followed three models.

  1. Utilize VC funding to increase your user base, then monetize them later. (If you do this, you won't kill Facebook; instead, Facebook will become you.)

  2. Users must pay to utilize it. (This causes a huge bottleneck and slows the required quick expansion, preventing it from seeming like a true social network.)

  3. Make it a volunteer-run, open-source endeavor that is free. (This typically denotes that something is cumbersome, difficult to operate, and is only for techies.)

Wikipedia is a fourth way.

Wikipedia is one of the most popular websites and a charity. No ads. Donations support them.

A Facebook-killer managed by a good team may gather millions (from affluent contributors and the crowd) for their initial phase of development. Then it might sustain on regular donations, ethical transactions (e.g. fees on commerce, business sites, etc.), and government grants/subsidies (since it would essentially be a public utility).

When you're not aiming to make investors rich, it's remarkable how little money you need.

If you want to build a Facebook competitor, follow these tips:

  1. Drop the lean startup philosophy. Wait until you have a finished product before launching. Build it, thoroughly test it for bugs, and then release it.

  2. Delay innovating. Wait till millions of people have switched before introducing your great new features. Make it nearly identical for now.

  3. Spend money climbing. Make sure that guests can arrive as soon as they are invited. Never keep them waiting. Make things easy for them.

  4. Make it accessible to all. Even if doing so renders it less philosophically pure, it shouldn't require technical expertise to utilize.

  5. Constitute a nonprofit. Additionally, develop community ownership structures. Profit maximization is not the only strategy for preserving valued assets.

Last thoughts

Nobody has killed Facebook, but Facebook is killing itself.

The startup is burying the newsfeed to become a TikTok clone. Meta itself seems to be ditching the platform for the metaverse.

I wish I was happy, but I'm not. I miss (understandably) removed friends' postings and remarks. It could be a ghost town in a few years. My dance moves aren't TikTok-worthy.

Who will lead? It's time to develop a social network for the people.

Greetings if you're working on it. I'm not a company founder, but I like to help hard-working folks.

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Jano le Roux

Jano le Roux

3 years ago

My Top 11 Tools For Building A Modern Startup, With A Free Plan

The best free tools are probably unknown to you.

Webflow

Modern startups are easy to build.

Start with free tools.

Let’s go.

Web development — Webflow

Code-free HTML, CSS, and JS.

Webflow isn't like Squarespace, Wix, or Shopify.

It's a super-fast no-code tool for professionals to construct complex, highly-responsive websites and landing pages.

Webflow can help you add animations like those on Apple's website to your own site.

I made the jump from WordPress a few years ago and it changed my life.

No damn plugins. No damn errors. No damn updates.

The best, you can get started on Webflow for free.

Data tracking — Airtable

Spreadsheet wings.

Airtable combines spreadsheet flexibility with database power without code.

  • Airtable is modern.

  • Airtable has modularity.

  • Scaling Airtable is simple.

Airtable, one of the most adaptable solutions on this list, is perfect for client data management.

Clients choose customized service packages. Airtable consolidates data so you can automate procedures like invoice management and focus on your strengths.

Airtable connects with so many tools that rarely creates headaches. Airtable scales when you do.

Airtable's flexibility makes it a potential backend database.

Design — Figma

Better, faster, easier user interface design.

Figma rocks!

  • It’s fast.

  • It's free.

  • It's adaptable

First, design in Figma.

Iterate.

Export development assets.

Figma lets you add more team members as your company grows to work on each iteration simultaneously.

Figma is web-based, so you don't need a powerful PC or Mac to start.

Task management — Trello

Unclock jobs.

Tacky and terrifying task management products abound. Trello isn’t.

Those that follow Marie Kondo will appreciate Trello.

  • Everything is clean.

  • Nothing is complicated.

  • Everything has a place.

Compared to other task management solutions, Trello is limited. And that’s good. Too many buttons lead to too many decisions lead to too many hours wasted.

Trello is a must for teamwork.

Domain email — Zoho

Free domain email hosting.

Professional email is essential for startups. People relied on monthly payments for too long. Nope.

Zoho offers 5 free professional emails.

It doesn't have Google's UI, but it works.

VPN — Proton VPN

Fast Swiss VPN protects your data and privacy.

Proton VPN is secure.

  • Proton doesn't record any data.

  • Proton is based in Switzerland.

Swiss privacy regulation is among the most strict in the world, therefore user data are protected. Switzerland isn't a 14 eye country.

Journalists and activists trust Proton to secure their identities while accessing and sharing information authoritarian governments don't want them to access.

Web host — Netlify

Free fast web hosting.

Netlify is a scalable platform that combines your favorite tools and APIs to develop high-performance sites, stores, and apps through GitHub.

Serverless functions and environment variables preserve API keys.

Netlify's free tier is unmissable.

  • 100GB of free monthly bandwidth.

  • Free 125k serverless operations per website each month.

Database — MongoDB

Create a fast, scalable database.

MongoDB is for small and large databases. It's a fast and inexpensive database.

  • Free for the first million reads.

  • Then, for each million reads, you must pay $0.10.

MongoDB's free plan has:

  • Encryption from end to end

  • Continual authentication

  • field-level client-side encryption

If you have a large database, you can easily connect MongoDB to Webflow to bypass CMS limits.

Automation — Zapier

Time-saving tip: automate repetitive chores.

Zapier simplifies life.

Zapier syncs and connects your favorite apps to do impossibly awesome things.

If your online store is connected to Zapier, a customer's purchase can trigger a number of automated actions, such as:

  1. The customer is being added to an email chain.

  2. Put the information in your Airtable.

  3. Send a pre-programmed postcard to the customer.

  4. Alexa, set the color of your smart lights to purple.

Zapier scales when you do.

Email & SMS marketing — Omnisend

Email and SMS marketing campaigns.

Omnisend

This is an excellent Mailchimp option for magical emails. Omnisend's processes simplify email automation.

I love the interface's cleanliness.

Omnisend's free tier includes web push notifications.

Send up to:

  • 500 emails per month

  • 60 maximum SMSs

  • 500 Web Push Maximum

Forms and surveys — Tally

Create flexible forms that people enjoy.

Typeform is clean but restricting. Sometimes you need to add many questions. Tally's needed sometimes.

Tally is flexible and cheaper than Typeform.

99% of Tally's features are free and unrestricted, including:

  • Unlimited forms

  • Countless submissions

  • Collect payments

  • File upload

Tally lets you examine what individuals contributed to forms before submitting them to see where they get stuck.

Airtable and Zapier connectors automate things further. If you pay, you can apply custom CSS to fit your brand.

See.

Free tools are the greatest.

Let's use them to launch a startup.

Abhimanyu Bhargava

Abhimanyu Bhargava

3 years ago

VeeFriends Series 2: The Biggest NFT Opportunity Ever

VeeFriends is one NFT project I'm sure will last.

I believe in blockchain technology and JPEGs, aka NFTs. NFTs aren't JPEGs. It's not as it seems.

Gary Vaynerchuk is leading the pack with his new NFT project VeeFriends, I wrote a year ago. I was spot-on. It's the most innovative project I've seen.

Since its minting in May 2021, it has given its holders enormous value, most notably the first edition of VeeCon, a multi-day superconference featuring iconic and emerging leaders in NFTs and Popular Culture. First-of-its-kind NFT-ticketed Web3 conference to build friendships, share ideas, and learn together.

VeeFriends holders got free VeeCon NFT tickets. Attendees heard iconic keynote speeches, innovative talks, panels, and Q&A sessions.

It was a unique conference that most of us, including me, are looking forward to in 2023. The lineup was epic, and it allowed many to network in new ways. Really memorable learning. Here are a couple of gratitude posts from the attendees.

VeeFriends Series 2

This article explains VeeFriends if you're still confused.

GaryVee's hand-drawn doodles have evolved into wonderful characters. The characters' poses and backgrounds bring the VeeFriends IP to life.

Yes, this is the second edition of VeeFriends, and at current prices, it's one of the best NFT opportunities in years. If you have the funds and risk appetite to invest in NFTs, VeeFriends Series 2 is worth every penny. Even if you can't invest, learn from their journey.

1. Art Is the Start

Many critics say VeeFriends artwork is below average and not by GaryVee. Art is often the key to future success.

Let's look at one of the first Mickey Mouse drawings. No one would have guessed that this would become one of the most beloved animated short film characters. In Walt Before Mickey, Walt Disney's original mouse Mortimer was less refined.

First came a mouse...

These sketches evolved into Steamboat Willie, Disney's first animated short film.

Fred Moore redesigned the character artwork into what we saw in cartoons as kids. Mickey Mouse's history is here.

Looking at how different cartoon characters have evolved and gained popularity over decades, I believe Series 2 characters like Self-Aware Hare, Kind Kudu, and Patient Pig can do the same.

GaryVee captures this journey on the blockchain and lets early supporters become part of history. Time will tell if it rivals Disney, Pokemon, or Star Wars. Gary has been vocal about this vision.

2. VeeFriends is Intellectual Property for the Coming Generations

Most of us grew up watching cartoons, playing with toys, cards, and video games. Our interactions with fictional characters and the stories we hear shape us.

GaryVee is slowly curating an experience for the next generation with animated videos, card games, merchandise, toys, and more.

VeeFriends UNO, a collaboration with Mattel Creations, features 17 VeeFriends characters.

VeeFriends and Zerocool recently released Trading Cards featuring all 268 Series 1 characters and 15 new ones. Another way to build VeeFriends' collectibles brand.

At Veecon, all the characters were collectible toys. Something will soon emerge.

Kids and adults alike enjoy the YouTube channel's animated shorts and VeeFriends Tunes. Here's a song by the holder's Optimistic Otter-loving daughter.

This VeeFriends story is only the beginning. I'm looking forward to animated short film series, coloring books, streetwear, candy, toys, physical collectibles, and other forms of VeeFriends IP.

3. Veefriends will always provide utilities

Smart contracts can be updated at any time and authenticated on a ledger.

VeeFriends Series 2 gives no promise of any utility whatsoever. GaryVee released no project roadmap. In the first few months after launch, many owners of specific characters or scenes received utilities.

Every benefit or perk you receive helps promote the VeeFriends brand.

Recent partnerships are listed below.

  • MaryRuth's Multivitamin Gummies

  • Productive Puffin holders from VeeFriends x Primitive

  • Pickleball Scene & Clown Holders Only

Pickleball & Competitive Clown Exclusive experience, anteater multivitamin gummies, and Puffin x Primitive merch

Considering the price of NFTs, it may not seem like much. It's just the beginning; you never know what the future holds. No other NFT project offers such diverse, ongoing benefits.

4. Garyvee's team is ready

Gary Vaynerchuk's team and record are undisputed. He's a serial entrepreneur and the Chairman & CEO of VaynerX, which includes VaynerMedia, VaynerCommerce, One37pm, and The Sasha Group.

Gary founded VaynerSports, Resy, and Empathy Wines. He's a Candy Digital Board Member, VCR Group Co-Founder, ArtOfficial Co-Founder, and VeeFriends Creator & CEO. Gary was recently named one of Fortune's Top 50 NFT Influencers.

Gary Vayenerchuk aka GaryVee

Gary documents his daily life as a CEO on social media, which has 34 million followers and 272 million monthly views. GaryVee Audio Experience is a top podcast. He's a five-time New York Times best-seller and sought-after speaker.

Gary can observe consumer behavior to predict trends. He understood these trends early and pioneered them.

  • 1997 — Realized e-potential commerce's and started winelibrary.com. In five years, he grew his father's wine business from $3M to $60M.

  • 2006 — Realized content marketing's potential and started Wine Library on YouTube. TV

  • 2009 — Estimated social media's potential (Web2) and invested in Facebook, Twitter, and Tumblr.

  • 2014: Ethereum and Bitcoin investments

  • 2021 — Believed in NFTs and Web3 enough to launch VeeFriends

GaryVee isn't all of VeeFriends. Andy Krainak, Dave DeRosa, Adam Ripps, Tyler Dowdle, and others work tirelessly to make VeeFriends a success.

GaryVee has said he'll let other businesses fail but not VeeFriends. We're just beginning his 40-year vision.

I have more confidence than ever in a company with a strong foundation and team.

5. Humans die, but characters live forever

What if GaryVee dies or can't work?

A writer's books can immortalize them. As long as their books exist, their words are immortal. Socrates, Hemingway, Aristotle, Twain, Fitzgerald, and others have become immortal.

Everyone knows Vincent Van Gogh's The Starry Night.

We all love reading and watching Peter Parker, Thor, or Jessica Jones. Their behavior inspires us. Stan Lee's message and stories live on despite his death.

GaryVee represents VeeFriends. Creating characters to communicate ensures that the message reaches even those who don't listen.

Gary wants his values and messages to be omnipresent in 268 characters. Messengers die, but their messages live on.

Gary envisions VeeFriends creating timeless stories and experiences. Ten years from now, maybe every kid will sing Patient Pig.

6. I love the intent.

Gary planned to create Workplace Warriors three years ago when he began designing Patient Panda, Accountable Ant, and Empathy elephant. The project stalled. When NFTs came along, he knew.

Gary wanted to create characters with traits he values, such as accountability, empathy, patience, kindness, and self-awareness. He wants future generations to find these traits cool. He hopes one or more of his characters will become pop culture icons.

These emotional skills aren't taught in schools or colleges, but they're crucial for business and life success. I love that someone is teaching this at scale.

In the end, intent matters.

Humans Are Collectors

Buy and collect things to communicate. Since the 1700s. Medieval people formed communities around hidden metals and stones. Many people still collect stamps and coins, and luxury and fashion are multi-trillion dollar industries. We're collectors.

The early 2020s NFTs will be remembered in the future. VeeFriends will define a cultural and technological shift in this era. VeeFriends Series 1 is the original hand-drawn art, but it's expensive. VeeFriends Series 2 is a once-in-a-lifetime opportunity at $1,000.

If you are new to NFTs, check out How to Buy a Non Fungible Token (NFT) For Beginners


This is a non-commercial article. Not financial or legal advice. Information isn't always accurate. Before making important financial decisions, consult a pro or do your own research.


This post is a summary. Read the full article here

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