# DeaMau5’s PIXELYNX and Beatport Launch Festival NFTs
Pixelynx, a music metaverse gaming platform, has teamed up with Beatport, an online music retailer focusing in electronic music, to establish a Synth Heads non-fungible token (NFT) Collection.
Richie Hawtin, aka Deadmau5, and Joel Zimmerman, nicknamed Pixelynx, have invented a new music metaverse game platform called Pixelynx. In January 2022, they released their first Beatport NFT drop, which saw 3,030 generative NFTs sell out in seconds.
The limited edition Synth Heads NFTs will be released in collaboration with Junction 2, the largest UK techno festival, and having one will grant fans special access tickets and experiences at the London-based festival.
Membership in the Synth Head community, day passes to the Junction 2 Festival 2022, Junction 2 and Beatport apparel, special vinyl releases, and continued access to future ticket drops are just a few of the experiences available.
Five lucky NFT holders will also receive a Golden Ticket, which includes access to a backstage artist bar and tickets to Junction 2's next large-scale London event this summer, in addition to full festival entrance for both days.
The Junction 2 festival will take place at Trent Park in London on June 18th and 19th, and will feature performances from Four Tet, Dixon, Amelie Lens, Robert Hood, and a slew of other artists. Holders of the original Synth Head NFT will be granted admission to the festival's guestlist as well as line-jumping privileges.
The new Synth Heads NFTs collection contain 300 NFTs.
NFTs that provide IRL utility are in high demand.
The benefits of NFT drops related to In Real Life (IRL) utility aren't limited to Beatport and Pixelynx.
Coachella, a well-known music event, recently partnered with cryptocurrency exchange FTX to offer free NFTs to 2022 pass holders. Access to a dedicated entry lane, a meal and beverage pass, and limited-edition merchandise were all included with the NFTs.
Coachella also has its own NFT store on the Solana blockchain, where fans can buy Coachella NFTs and digital treasures that unlock exclusive on-site experiences, physical objects, lifetime festival passes, and "future adventures."
Individual artists and performers have begun taking advantage of NFT technology outside of large music festivals like Coachella.
DJ Tisto has revealed that he would release a VIP NFT for his upcoming "Eagle" collection during the EDC festival in Las Vegas in 2022. This NFT, dubbed "All Access Eagle," gives collectors the best chance to get NFTs from his first drop, as well as unique access to the music "Repeat It."
NFTs are one-of-a-kind digital assets that can be verified, purchased, sold, and traded on blockchains, opening up new possibilities for artists and businesses alike. Time will tell whether Beatport and Pixelynx's Synth Head NFT collection will be successful, but if it's anything like the first release, it's a safe bet.
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.

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?
- 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.
- 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.
- 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.
Read original post here

nft now
3 years ago
Instagram NFTs Are Here… How does this affect artists?
Instagram (IG) is officially joining NFT. With the debut of new in-app NFT functionalities, influential producers can interact with blockchain tech on the social media platform.
Meta unveiled intentions for an Instagram NFT marketplace in March, but these latest capabilities focus more on content sharing than commerce. And why shouldn’t they? IG's entry into the NFT market is overdue, given that Twitter and Discord are NFT hotspots.
The NFT marketplace/Web3 social media race has continued to expand, with the expected Coinbase NFT Beta now live and blazing a trail through the NFT ecosystem.
IG's focus is on visual art. It's unlike any NFT marketplace or platform. IG NFTs and artists: what's the deal? Let’s take a look.
What are Instagram’s NFT features anyways?
As said, not everyone has Instagram's new features. 16 artists, NFT makers, and collectors can now post NFTs on IG by integrating third-party digital wallets (like Rainbow or MetaMask) in-app. IG doesn't charge to publish or share digital collectibles.
NFTs displayed on the app have a "shimmer" aesthetic effect. NFT posts also have a "digital collectable" badge that lists metadata such as the creator and/or owner, the platform it was created on, a brief description, and a blockchain identification.
Meta's social media NFTs have launched on Instagram, but the company is also preparing to roll out digital collectibles on Facebook, with more on the way for IG. Currently, only Ethereum and Polygon are supported, but Flow and Solana will be added soon.
How will artists use these new features?
Artists are publishing NFTs they developed or own on IG by linking third-party digital wallets. These features have no NFT trading aspects built-in, but are aimed to let authors share NFTs with IG audiences.
Creators, like IG-native aerial/street photographer Natalie Amrossi (@misshattan), are discovering novel uses for IG NFTs.
Amrossi chose to not only upload his own NFTs but also encourage other artists in the field. "That's the beauty of connecting your wallet and sharing NFTs. It's not just what you make, but also what you accumulate."
Amrossi has been producing and posting Instagram art for years. With IG's NFT features, she can understand Instagram's importance in supporting artists.
Web2 offered Amrossi the tools to become an artist and make a life. "Before 'influencer' existed, I was just making art. Instagram helped me reach so many individuals and brands, giving me a living.
Even artists without millions of viewers are encouraged to share NFTs on IG. Wilson, a relatively new name in the NFT space, seems to have already gone above and beyond the scope of these new IG features. By releasing "Losing My Mind" via IG NFT posts, she has evaded the lack of IG NFT commerce by using her network to market her multi-piece collection.
"'Losing My Mind' is a long-running photo series. Wilson was preparing to release it as NFTs before IG approached him, so it was a perfect match.
Wilson says the series is about Black feminine figures and media depiction. Respectable effort, given POC artists have been underrepresented in NFT so far.
“Over the past year, I've had mental health concerns that made my emotions so severe it was impossible to function in daily life, therefore that prompted this photo series. Every Wednesday and Friday for three weeks, I'll release a new Meta photo for sale.
Wilson hopes these new IG capabilities will help develop a connection between the NFT community and other internet subcultures that thrive on Instagram.
“NFTs can look scary as an outsider, but seeing them on your daily IG feed makes it less foreign,” adds Wilson. I think Instagram might become a hub for NFT aficionados, making them more accessible to artists and collectors.
What does it all mean for the NFT space?
Meta's NFT and metaverse activities will continue to impact Instagram's NFT ecosystem. Many think it will be for the better, as IG NFT frauds are another problem hurting the NFT industry.
IG's new NFT features seem similar to Twitter's PFP NFT verifications, but Instagram's tools should help cut down on scams as users can now verify the creation and ownership of whole NFT collections included in IG posts.
Given the number of visual artists and NFT creators on IG, it might become another hub for NFT fans, as Wilson noted. If this happens, it raises questions about Instagram success. Will artists be incentivized to distribute NFTs? Or will those with a large fanbase dominate?
Elise Swopes (@swopes) believes these new features should benefit smaller artists. Swopes was one of the first profiles placed to Instagram's original suggested user list in 2012.
Swopes says she wants IG to be a magnet for discovery and understands the value of NFT artists and producers.
"I'd love to see IG become a focus of discovery for everyone, not just the Beeples and Apes and PFPs. That's terrific for them, but [IG NFT features] are more about using new technology to promote emerging artists, Swopes added.
“Especially music artists. It's everywhere. Dancers, writers, painters, sculptors, musicians. My element isn't just for digital artists; it can be anything. I'm delighted to witness people's creativity."
Swopes, Wilson, and Amrossi all believe IG's new features can help smaller artists. It remains to be seen how these new features will effect the NFT ecosystem once unlocked for the rest of the IG NFT community, but we will likely see more social media NFT integrations in the months and years ahead.
Read the full article here
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Tim Denning
3 years ago
One of the biggest publishers in the world offered me a book deal, but I don't feel deserving of it.
My ego is so huge it won't fit through the door.
I don't know how I feel about it. I should be excited. Many of you have this exact dream to publish a book with a well-known book publisher and get a juicy advance.
Let me dissect how I'm thinking about it to help you.
How it happened
An email comes in. A generic "can we put a backlink on your website and get a freebie" email.
Almost deleted it.
Then I noticed the logo. It seemed shady. I found the URL. Check. I searched the employee's LinkedIn. Legit. I avoided middlemen. Check.
Mixed feelings. LinkedIn hasn't valued my writing for years. I'm just a guy in an unironed t-shirt whose content they sell advertising against.
They get big dollars. I get $0 and a few likes, plus some email subscribers.
Still, I felt adrenaline for hours.
I texted a few friends to see how they felt. I wrapped them.
Messages like "No shocker. You're entertaining online." I didn't like praises, so I blushed.
The thrill faded after hours. Who knows?
Most authors desire this chance.
"You entitled piece of crap, Denning!"
You may think so. Okay. My job is to stand on the internet and get bananas thrown at me.
I approached writing backwards. More important than a book deal was a social media audience converted to an email list.
Romantic authors think backward. They hope a fantastic book will land them a deal and an audience.
Rarely occurs. So I never pursued it. It's like permission-seeking or the lottery.
Not being a professional writer, I've never written a good book. I post online for fun and to express my opinions.
Writing is therapeutic. I overcome mental illness and rebuilt my life this way. Without blogging, I'd be dead.
I've always dreamed of staying alive and doing something I love, not getting a book contract. Writing is my passion. I'm a winner without a book deal.
Why I was given a book deal
You may assume I received a book contract because of my views or follows. Nope.
They gave me a deal because they like my writing style. I've heard this for eight years.
Several authors agree. One asked me to improve their writer's voice.
Takeaway: highlight your writer's voice.
What if they discover I'm writing incompetently?
An edited book is published. It's edited.
I need to master writing mechanics, thus this concerns me. I need help with commas and sentence construction.
I must learn verb, noun, and adjective. Seriously.
Writing a book may reveal my imposter status to a famous publisher. Imagine the email
"It happened again. He doesn't even know how to spell. He thinks 'less' is the correct word, not 'fewer.' Are you sure we should publish his book?"
Fears stink.
I'm capable of blogging. Even listicles. So what?
Writing for a major publisher feels advanced.
I only blog. I'm good at listicles. Digital media executives have criticized me for this.
It is allegedly clickbait.
Or it is following trends.
Alternately, growth hacking.
Never. I learned copywriting to improve my writing.
Apple, Amazon, and Tesla utilize copywriting to woo customers. Whoever thinks otherwise is the wisest person in the room.
Old-schoolers loathe copywriters.
Their novels sell nothing.
They assume their elitist version of writing is better and that the TikTok generation will invest time in random writing with no subheadings and massive walls of text they can't read on their phones.
I'm terrified of book proposals.
My friend's book proposal suggestion was contradictory and made no sense.
They told him to compose another genre. This book got three Amazon reviews. Is that a good model?
The process disappointed him. I've heard other book proposal horror stories. Tim Ferriss' book "The 4-Hour Workweek" was criticized.
Because he has thick skin, his book came out. He wouldn't be known without that.
I hate book proposals.
An ongoing commitment
Writing a book is time-consuming.
I appreciate time most. I want to focus on my daughter for the next few years. I can't recreate her childhood because of a book.
No idea how parents balance kids' goals.
My silly face in a bookstore. Really?
Genuine thought.
I don't want my face in bookstores. I fear fame. I prefer anonymity.
I want to purchase a property in a bad Australian area, then piss off and play drums. Is bookselling worth it?
Are there even bookstores anymore?
(Except for Ryan Holiday's legendary Painted Porch Bookshop in Texas.)
What's most important about books
Many were duped.
Tweets and TikTok hopscotch vids are their future. Short-form content creates devoted audiences that buy newsletter subscriptions.
Books=depth.
Depth wins (if you can get people to buy your book). Creating a book will strengthen my reader relationships.
It's cheaper than my classes, so more people can benefit from my life lessons.
A deeper justification for writing a book
Mind wandered.
If I write this book, my daughter will follow it. "Look what you can do, love, when you ignore critics."
That's my favorite.
I'll be her best leader and teacher. If her dad can accomplish this, she can too.
My kid can read my book when I'm gone to remember her loving father.
Last paragraph made me cry.
The positive
This book thing might make me sound like Karen.
The upside is... Building in public, like I have with online writing, attracts the right people.
Proof-of-work over proposals, beautiful words, or huge aspirations. If you want a book deal, try writing online instead of the old manner.
Next steps
No idea.
I'm a rural Aussie. Writing a book in the big city is intimidating. Will I do it? Lots to think about. Right now, some level of reflection and gratitude feels most appropriate.
Sometimes when you don't feel worthy, it gives you the greatest lessons. That's how I feel about getting offered this book deal.
Perhaps you can relate.
Sam Hickmann
3 years ago
What is headline inflation?
Headline inflation is the raw Consumer price index (CPI) reported monthly by the Bureau of labour statistics (BLS). CPI measures inflation by calculating the cost of a fixed basket of goods. The CPI uses a base year to index the current year's prices.
Explaining Inflation
As it includes all aspects of an economy that experience inflation, headline inflation is not adjusted to remove volatile figures. Headline inflation is often linked to cost-of-living changes, which is useful for consumers.
The headline figure doesn't account for seasonality or volatile food and energy prices, which are removed from the core CPI. Headline inflation is usually annualized, so a monthly headline figure of 4% inflation would equal 4% inflation for the year if repeated for 12 months. Top-line inflation is compared year-over-year.
Inflation's downsides
Inflation erodes future dollar values, can stifle economic growth, and can raise interest rates. Core inflation is often considered a better metric than headline inflation. Investors and economists use headline and core results to set growth forecasts and monetary policy.
Core Inflation
Core inflation removes volatile CPI components that can distort the headline number. Food and energy costs are commonly removed. Environmental shifts that affect crop growth can affect food prices outside of the economy. Political dissent can affect energy costs, such as oil production.
From 1957 to 2018, the U.S. averaged 3.64 percent core inflation. In June 1980, the rate reached 13.60%. May 1957 had 0% inflation. The Fed's core inflation target for 2022 is 3%.
Central bank:
A central bank has privileged control over a nation's or group's money and credit. Modern central banks are responsible for monetary policy and bank regulation. Central banks are anti-competitive and non-market-based. Many central banks are not government agencies and are therefore considered politically independent. Even if a central bank isn't government-owned, its privileges are protected by law. A central bank's legal monopoly status gives it the right to issue banknotes and cash. Private commercial banks can only issue demand deposits.
What are living costs?
The cost of living is the amount needed to cover housing, food, taxes, and healthcare in a certain place and time. Cost of living is used to compare the cost of living between cities and is tied to wages. If expenses are higher in a city like New York, salaries must be higher so people can live there.
What's U.S. bureau of labor statistics?
BLS collects and distributes economic and labor market data about the U.S. Its reports include the CPI and PPI, both important inflation measures.

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