More on NFTs & Art
Jayden Levitt
3 years ago
How to Explain NFTs to Your Grandmother, in Simple Terms
In simple terms, you probably don’t.
But try. Grandma didn't grow up with Facebook, but she eventually joined.
Perhaps the fear of being isolated outweighed the discomfort of learning the technology.
Grandmas are Facebook likers, sharers, and commenters.
There’s no stopping her.
Not even NFTs. Web3 is currently very complex.
It's difficult to explain what NFTs are, how they work, and why we might use them.
Three explanations.
1. Everything will be ours to own, both physically and digitally.
Why own something you can't touch? What's the point?
Blockchain technology proves digital ownership.
Untouchables need ownership proof. What?
Digital assets reduce friction, save time, and are better for the environment than physical goods.
Many valuable things are intangible. Feeling like your favorite brands. You'll pay obscene prices for clothing that costs pennies.
Secondly, NFTs Are Contracts. Agreements Have Value.
Blockchain technology will replace all contracts and intermediaries.
Every insurance contract, deed, marriage certificate, work contract, plane ticket, concert ticket, or sports event is likely an NFT.
We all have public wallets, like Grandma's Facebook page.
3. Your NFT Purchases Will Be Visible To Everyone.
Everyone can see your public wallet. What you buy says more about you than what you post online.
NFTs issued double as marketing collateral when seen on social media.
While I doubt Grandma knows who Snoop Dog is, imagine him or another famous person holding your NFT in his public wallet and the attention that could bring to you, your company, or brand.
This Technical Section Is For You
The NFT is a contract; its founders can add value through access, events, tuition, and possibly royalties.
Imagine Elon Musk releasing an NFT to his network. Or yearly business consultations for three years.
Christ-alive.
It's worth millions.
These determine their value.
No unsuspecting schmuck willing to buy your hot potato at zero. That's the trend, though.
Overpriced NFTs for low-effort projects created a bubble that has burst.
During a market bubble, you can make money by buying overvalued assets and selling them later for a profit, according to the Greater Fool Theory.
People are struggling. Some are ruined by collateralized loans and the gold rush.
Finances are ruined.
It's uncomfortable.
The same happened in 2018, during the ICO crash or in 1999/2000 when the dot com bubble burst. But the underlying technology hasn’t gone away.
shivsak
3 years ago
A visual exploration of the REAL use cases for NFTs in the Future
In this essay, I studied REAL NFT use examples and their potential uses.
Knowledge of the Hype Cycle
Gartner's Hype Cycle.
It proposes 5 phases for disruptive technology.
1. Technology Trigger: the emergence of potentially disruptive technology.
2. Peak of Inflated Expectations: Early publicity creates hype. (Ex: 2021 Bubble)
3. Trough of Disillusionment: Early projects fail to deliver on promises and the public loses interest. I suspect NFTs are somewhere around this trough of disillusionment now.
4. Enlightenment slope: The tech shows successful use cases.
5. Plateau of Productivity: Mainstream adoption has arrived and broader market applications have proven themselves. Here’s a more detailed visual of the Gartner Hype Cycle from Wikipedia.
In the speculative NFT bubble of 2021, @beeple sold Everydays: the First 5000 Days for $69 MILLION in 2021's NFT bubble.
@nbatopshot sold millions in video collectibles.
This is when expectations peaked.
Let's examine NFTs' real-world applications.
Watch this video if you're unfamiliar with NFTs.
Online Art
Most people think NFTs are rich people buying worthless JPEGs and MP4s.
Digital artwork and collectibles are revolutionary for creators and enthusiasts.
NFT Profile Pictures
You might also have seen NFT profile pictures on Twitter.
My profile picture is an NFT I coined with @skogards factoria app, which helps me avoid bogus accounts.
Profile pictures are a good beginning point because they're unique and clearly yours.
NFTs are a way to represent proof-of-ownership. It’s easier to prove ownership of digital assets than physical assets, which is why artwork and pfps are the first use cases.
They can do much more.
NFTs can represent anything with a unique owner and digital ownership certificate. Domains and usernames.
Usernames & Domains
@unstoppableweb, @ensdomains, @rarible sell NFT domains.
NFT domains are transferable, which is a benefit.
Godaddy and other web2 providers have difficult-to-transfer domains. Domains are often leased instead of purchased.
Tickets
NFTs can also represent concert tickets and event passes.
There's a limited number, and entry requires proof.
NFTs can eliminate the problem of forgery and make it easy to verify authenticity and ownership.
NFT tickets can be traded on the secondary market, which allows for:
marketplaces that are uniform and offer the seller and buyer security (currently, tickets are traded on inefficient markets like FB & craigslist)
unbiased pricing
Payment of royalties to the creator
4. Historical ticket ownership data implies performers can airdrop future passes, discounts, etc.
5. NFT passes can be a fandom badge.
The $30B+ online tickets business is increasing fast.
NFT-based ticketing projects:
Gaming Assets
NFTs also help in-game assets.
Imagine someone spending five years collecting a rare in-game blade, then outgrowing or quitting the game. Gamers value that collectible.
The gaming industry is expected to make $200 BILLION in revenue this year, a significant portion of which comes from in-game purchases.
Royalties on secondary market trading of gaming assets encourage gaming businesses to develop NFT-based ecosystems.
Digital assets are the start. On-chain NFTs can represent real-world assets effectively.
Real estate has a unique owner and requires ownership confirmation.
Real Estate
Tokenizing property has many benefits.
1. Can be fractionalized to increase access, liquidity
2. Can be collateralized to increase capital efficiency and access to loans backed by an on-chain asset
3. Allows investors to diversify or make bets on specific neighborhoods, towns or cities +++
I've written about this thought exercise before.
I made an animated video explaining this.
We've just explored NFTs for transferable assets. But what about non-transferrable NFTs?
SBTs are Soul-Bound Tokens. Vitalik Buterin (Ethereum co-founder) blogged about this.
NFTs are basically verifiable digital certificates.
Diplomas & Degrees
That fits Degrees & Diplomas. These shouldn't be marketable, thus they can be non-transferable SBTs.
Anyone can verify the legitimacy of on-chain credentials, degrees, abilities, and achievements.
The same goes for other awards.
For example, LinkedIn could give you a verified checkmark for your degree or skills.
Authenticity Protection
NFTs can also safeguard against counterfeiting.
Counterfeiting is the largest criminal enterprise in the world, estimated to be $2 TRILLION a year and growing.
Anti-counterfeit tech is valuable.
This is one of @ORIGYNTech's projects.
Identity
Identity theft/verification is another real-world problem NFTs can handle.
In the US, 15 million+ citizens face identity theft every year, suffering damages of over $50 billion a year.
This isn't surprising considering all you need for US identity theft is a 9-digit number handed around in emails, documents, on the phone, etc.
Identity NFTs can fix this.
NFTs are one-of-a-kind and unforgeable.
NFTs offer a universal standard.
NFTs are simple to verify.
SBTs, or non-transferrable NFTs, are tied to a particular wallet.
In the event of wallet loss or theft, NFTs may be revoked.
This could be one of the biggest use cases for NFTs.
Imagine a global identity standard that is standardized across countries, cannot be forged or stolen, is digital, easy to verify, and protects your private details.
Since your identity is more than your government ID, you may have many NFTs.
@0xPolygon and @civickey are developing on-chain identity.
Memberships
NFTs can authenticate digital and physical memberships.
Voting
NFT IDs can verify votes.
If you remember 2020, you'll know why this is an issue.
Online voting's ease can boost turnout.
Informational property
NFTs can protect IP.
This can earn creators royalties.
NFTs have 2 important properties:
Verifiability IP ownership is unambiguously stated and publicly verified.
Platforms that enable authors to receive royalties on their IP can enter the market thanks to standardization.
Content Rights
Monetization without copyrighting = more opportunities for everyone.
This works well with the music.
Spotify and Apple Music pay creators very little.
Crowdfunding
Creators can crowdfund with NFTs.
NFTs can represent future royalties for investors.
This is particularly useful for fields where people who are not in the top 1% can’t make money. (Example: Professional sports players)
Mirror.xyz allows blog-based crowdfunding.
Financial NFTs
This introduces Financial NFTs (fNFTs). Unique financial contracts abound.
Examples:
a person's collection of assets (unique portfolio)
A loan contract that has been partially repaid with a lender
temporal tokens (ex: veCRV)
Legal Agreements
Not just financial contracts.
NFT can represent any legal contract or document.
Messages & Emails
What about other agreements? Verbal agreements through emails and messages are likewise unique, but they're easily lost and fabricated.
Health Records
Medical records or prescriptions are another types of documentation that has to be verified but isn't.
Medical NFT examples:
Immunization records
Covid test outcomes
Prescriptions
health issues that may affect one's identity
Observations made via health sensors
Existing systems of proof by paper / PDF have photoshop-risk.
I tried to include most use scenarios, but this is just the beginning.
NFTs have many innovative uses.
For example: @ShaanVP minted an NFT called “5 Minutes of Fame” 👇
Here are 2 Twitter threads about NFTs:
This piece of gold by @chriscantino
2. This conversation between @punk6529 and @RaoulGMI on @RealVision“The World According to @punk6529”
If you're wondering why NFTs are better than web2 databases for these use scenarios, see this Twitter thread I wrote:
If you liked this, please share it.
nft now
3 years ago
A Guide to VeeFriends and Series 2
VeeFriends is one of the most popular and unique NFT collections. VeeFriends launched around the same time as other PFP NFTs like Bored Ape Yacht Club.
Vaynerchuk (GaryVee) took a unique approach to his large-scale project, which has influenced the NFT ecosystem. GaryVee's VeeFriends is one of the most successful NFT membership use-cases, allowing him to build a community around his creative and business passions.
What is VeeFriends?
GaryVee's NFT collection, VeeFriends, was released on May 11, 2021. VeeFriends [Mini Drops], Book Games, and a forthcoming large-scale "Series 2" collection all stem from the initial drop of 10,255 tokens.
In "Series 1," there are G.O.O. tokens (Gary Originally Owned). GaryVee reserved 1,242 NFTs (over 12% of the supply) for his own collection, so only 9,013 were available at the Series 1 launch.
Each Series 1 token represents one of 268 human traits hand-drawn by Vaynerchuk. Gary Vee's NFTs offer owners incentives.
Who made VeeFriends?
Gary Vaynerchuk, AKA GaryVee, is influential in NFT. Vaynerchuk is the chairman of New York-based communications company VaynerX. Gary Vee, CEO of VaynerMedia, VaynerSports, and bestselling author, is worth $200 million.
GaryVee went from NFT collector to creator, launching VaynerNFT to help celebrities and brands.
Vaynerchuk's influence spans the NFT ecosystem as one of its most prolific voices. He's one of the most influential NFT figures, and his VeeFriends ecosystem keeps growing.
Vaynerchuk, a trend expert, thinks NFTs will be around for the rest of his life and VeeFriends will be a landmark project.
Why use VeeFriends NFTs?
The first VeeFriends collection has sold nearly $160 million via OpenSea. GaryVee insisted that the first 10,255 VeeFriends were just the beginning.
Book Games were announced to the VeeFriends community in August 2021. Mini Drops joined VeeFriends two months later.
Book Games
GaryVee's book "Twelve and a Half: Leveraging the Emotional Ingredients for Business Success" inspired Book Games. Even prior to the announcement Vaynerchuk had mapped out the utility of the book on an NFT scale. Book Games tied his book to the VeeFriends ecosystem and solidified its place in the collection.
GaryVee says Book Games is a layer 2 NFT project with 125,000 burnable tokens. Vaynerchuk's NFT fans were incentivized to buy as many copies of his new book as possible to receive NFT rewards later.
First, a bit about “layer 2.”
Layer 2 blockchain solutions help scale applications by routing transactions away from Ethereum Mainnet (layer 1). These solutions benefit from Mainnet's decentralized security model but increase transaction speed and reduce gas fees.
Polygon (integrated into OpenSea) and Immutable X are popular Ethereum layer 2 solutions. GaryVee chose Immutable X to reduce gas costs (transaction fees). Given the large supply of Book Games tokens, this decision will likely benefit the VeeFriends community, especially if the games run forever.
What's the strategy?
The VeeFriends patriarch announced on Aug. 27, 2021, that for every 12 books ordered during the Book Games promotion, customers would receive one NFT via airdrop. After nearly 100 days, GV sold over a million copies and announced that Book Games would go gamified on Jan. 10, 2022.
Immutable X's trading options make Book Games a "game." Book Games players can trade NFTs for other NFTs, sports cards, VeeCon tickets, and other prizes. Book Games can also whitelist other VeeFirends projects, which we'll cover in Series 2.
VeeFriends Mini Drops
GaryVee launched VeeFriends Mini Drops two months after Book Games, focusing on collaboration, scarcity, and the characters' "cultural longevity."
Spooky Vees, a collection of 31 1/1 Halloween-themed VeeFriends, was released on Halloween. First-come, first-served VeeFriend owners could claim these NFTs.
Mini Drops includes Gift Goat NFTs. By holding the Gift Goat VeeFriends character, collectors will receive 18 exclusive gifts curated by GaryVee and the team. Each gifting experience includes one physical gift and one NFT out of 555, to match the 555 Gift Goat tokens.
Gift Goat holders have gotten NFTs from Danny Cole (Creature World), Isaac "Drift" Wright (Where My Vans Go), Pop Wonder, and more.
GaryVee is poised to release the largest expansion of the VeeFriends and VaynerNFT ecosystem to date with VeeFriends Series 2.
VeeCon 101
By owning VeeFriends NFTs, collectors can join the VeeFriends community and attend VeeCon in 2022. The conference is only open to VeeCon NFT ticket holders (VeeFreinds + possibly more TBA) and will feature Beeple, Steve Aoki, and even Snoop Dogg.
The VeeFreinds floor in 2022 Q1 has remained at 16 ETH ($52,000), making VeeCon unattainable for most NFT enthusiasts. Why would someone spend that much crypto on a Minneapolis "superconference" ticket? Because of Gary Vaynerchuk.
Everything to know about VeeFriends Series 2
Vaynerchuk revealed in April 2022 that the VeeFriends ecosystem will grow by 55,555 NFTs after months of teasing.
With VeeFriends Series 2, each token will cost $995 USD in ETH, allowing NFT enthusiasts to join at a lower cost. The new series will be released on multiple dates in April.
Book Games NFT holders on the Friends List (whitelist) can mint Series 2 NFTs on April 12. Book Games holders have 32,000 NFTs.
VeeFriends Series 1 NFT holders can claim Series 2 NFTs on April 12. This allotment's supply is 10,255, like Series 1's.
On April 25, the public can buy 10,000 Series 2 NFTs. Unminted Friends List NFTs will be sold on this date, so this number may change.
The VeeFriends ecosystem will add 15 new characters (220 tokens each) on April 27. One character will be released per day for 15 days, and the only way to get one is to enter a daily raffle with Book Games tokens.
Series 2 NFTs won't give owners VeeCon access, but they will offer other benefits within the VaynerNFT ecosystem. Book Games and Series 2 will get new token burn mechanics in the upcoming drop.
Visit the VeeFriends blog for the latest collection info.
Where can you buy Gary Vee’s NFTs?
Need a VeeFriend NFT? Gary Vee recommends doing "50 hours of homework" before buying. OpenSea sells VeeFriends NFTs.
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Laura Sanders
3 years ago
Xenobots, tiny living machines, can duplicate themselves.
Strange and complex behavior of frog cell blobs
A xenobot “parent,” shaped like a hungry Pac-Man (shown in red false color), created an “offspring” xenobot (green sphere) by gathering loose frog cells in its opening.
Tiny “living machines” made of frog cells can make copies of themselves. This newly discovered renewal mechanism may help create self-renewing biological machines.
According to Kirstin Petersen, an electrical and computer engineer at Cornell University who studies groups of robots, “this is an extremely exciting breakthrough.” She says self-replicating robots are a big step toward human-free systems.
Researchers described the behavior of xenobots earlier this year (SN: 3/31/21). Small clumps of skin stem cells from frog embryos knitted themselves into small spheres and started moving. Cilia, or cellular extensions, powered the xenobots around their lab dishes.
The findings are published in the Proceedings of the National Academy of Sciences on Dec. 7. The xenobots can gather loose frog cells into spheres, which then form xenobots.
The researchers call this type of movement-induced reproduction kinematic self-replication. The study's coauthor, Douglas Blackiston of Tufts University in Medford, Massachusetts, and Harvard University, says this is typical. For example, sexual reproduction requires parental sperm and egg cells. Sometimes cells split or budded off from a parent.
“This is unique,” Blackiston says. These xenobots “find loose parts in the environment and cobble them together.” This second generation of xenobots can move like their parents, Blackiston says.
The researchers discovered that spheroid xenobots could only produce one more generation before dying out. The original xenobots' shape was predicted by an artificial intelligence program, allowing for four generations of replication.
A C shape, like an openmouthed Pac-Man, was predicted to be a more efficient progenitor. When improved xenobots were let loose in a dish, they began scooping up loose cells into their gaping “mouths,” forming more sphere-shaped bots (see image below). As many as 50 cells clumped together in the opening of a parent to form a mobile offspring. A xenobot is made up of 4,000–6,000 frog cells.
Petersen likes the Xenobots' small size. “The fact that they were able to do this at such a small scale just makes it even better,” she says. Miniature xenobots could sculpt tissues for implantation or deliver therapeutics inside the body.
Beyond the xenobots' potential jobs, the research advances an important science, says study coauthor and Tufts developmental biologist Michael Levin. The science of anticipating and controlling the outcomes of complex systems, he says.
“No one could have predicted this,” Levin says. “They regularly surprise us.” Researchers can use xenobots to test the unexpected. “This is about advancing the science of being less surprised,” Levin says.
Tim Denning
3 years ago
Bills are paid by your 9 to 5. 6 through 12 help you build money.
40 years pass. After 14 years of retirement, you die. Am I the only one who sees the problem?
I’m the Jedi master of escaping the rat race.
Not to impress. I know this works since I've tried it. Quitting a job to make money online is worse than Kim Kardashian's internet-burning advice.
Let me help you rethink the move from a career to online income to f*ck you money.
To understand why a job is a joke, do some life math.
Without a solid why, nothing makes sense.
The retirement age is 65. Our processed food consumption could shorten our 79-year average lifespan.
You spend 40 years working.
After 14 years of retirement, you die.
Am I alone in seeing the problem?
Life is too short to work a job forever, especially since most people hate theirs. After-hours skills are vital.
Money equals unrestricted power, f*ck you.
F*ck you money is the answer.
Jack Raines said it first. He says we can do anything with the money. Jack, a young rebel straight out of college, can travel and try new foods.
F*ck you money signifies not checking your bank account before buying.
F*ck you” money is pure, unadulterated freedom with no strings attached.
Jack claims you're rich when you rarely think about money.
Avoid confusion.
This doesn't imply you can buy a Lamborghini. It indicates your costs, income, lifestyle, and bank account are balanced.
Jack established an online portfolio while working for UPS in Atlanta, Georgia. So he gained boundless power.
The portion that many erroneously believe
Yes, you need internet abilities to make money, but they're not different from 9-5 talents.
Sahil Lavingia, Gumroad's creator, explains.
A job is a way to get paid to learn.
Mistreat your boss 9-5. Drain his skills. Defuse him. Love and leave him (eventually).
Find another employment if yours is hazardous. Pick an easy job. Make sure nothing sneaks into your 6-12 time slot.
The dumb game that makes you a sheep
A 9-5 job requires many job interviews throughout life.
You email your résumé to employers and apply for jobs through advertisements. This game makes you a sheep.
You're competing globally. Work-from-home makes the competition tougher. If you're not the cheapest, employers won't hire you.
After-hours online talents (say, 6 pm-12 pm) change the game. This graphic explains it better:
Online talents boost after-hours opportunities.
You go from wanting to be picked to picking yourself. More chances equal more money. Your f*ck you fund gets the extra cash.
A novel method of learning is essential.
College costs six figures and takes a lifetime to repay.
Informal learning is distinct. 6-12pm:
Observe the carefully controlled Twitter newsfeed.
Make use of Teachable and Gumroad's online courses.
Watch instructional YouTube videos
Look through the top Substack newsletters.
Informal learning is more effective because it's not obvious. It's fun to follow your curiosity and hobbies.
The majority of people lack one attitude. It's simple to learn.
One big impediment stands in the way of f*ck you money and time independence. So often.
Too many people plan after 6-12 hours. Dreaming. Big-thinkers. Strategically. They fill their calendar with meetings.
This is after-hours masturb*tion.
Sahil Bloom reminded me that a bias towards action will determine if this approach works for you.
The key isn't knowing what to do from 6-12 a.m. Trust yourself and develop abilities as you go. It's for building the parachute after you jump.
Sounds risky. We've eliminated the risk by finishing this process after hours while you work 9-5.
With no risk, you can have an I-don't-care attitude and still be successful.
When you choose to move forward, this occurs.
Once you try 9-5/6-12, you'll tell someone.
It's bad.
Few of us hang out with problem-solvers.
It's how much of society operates. So they make reasons so they can feel better about not giving you money.
Matthew Kobach told me chasing f*ck you money is easier with like-minded folks.
Without f*ck you money friends, loneliness will take over and you'll think you've messed up when you just need to keep going.
Steal this easy guideline
Let's act. No more fluffing and caressing.
1. Learn
If you detest your 9-5 talents or don't think they'll work online, get new ones. If you're skilled enough, continue.
Easlo recommends these skills:
Designer for Figma
Designer Canva
bubble creators
editor in Photoshop
Automation consultant for Zapier
Designer of Webflow
video editor Adobe
Ghostwriter for Twitter
Idea consultant
Artist in Blender Studio
2. Develop the ability
Every night from 6-12, apply the skill.
Practicing ghostwriting? Write someone's tweets for free. Do someone's website copy to learn copywriting. Get a website to the top of Google for a keyword to understand SEO.
Free practice is crucial. Your 9-5 pays the money, so work for free.
3. Take off stealthily like a badass
Another mistake. Sell to few. Don't be the best. Don't claim expertise.
Sell your new expertise to others behind you.
Two ways:
Using a digital good
By providing a service,
Point 1 also includes digital service examples. Digital products include eBooks, communities, courses, ad-supported podcasts, and templates. It's easy. Your 9-5 job involves one of these.
Take ideas from work.
Why? They'll steal your time for profit.
4. Iterate while feeling awful
First-time launches always fail. You'll feel terrible. Okay. Remember your 9-5?
Find improvements. Ask free and paying consumers what worked.
Multiple relaunches, each 1% better.
5. Discover more
Never stop learning. Improve your skill. Add a relevant skill. Learn copywriting if you write online.
After-hours students earn the most.
6. Continue
Repetition is key.
7. Make this one small change.
Consistently. The 6-12 momentum won't make you rich in 30 days; that's success p*rn.
Consistency helps wage slaves become f*ck you money. Most people can't switch between the two.
Putting everything together
It's easy. You're probably already doing some.
This formula explains why, how, and what to do. It's a 5th-grade-friendly blueprint. Good.
Reduce financial risk with your 9-to-5. Replace Netflix with 6-12 money-making talents.
Life is short; do whatever you want. Today.
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