Katrina Paulson

Katrina Paulson

1 year ago

Dehumanization Against Anthropomorphization

More on Science

Daniel Clery

1 year ago

Twisted device investigates fusion alternatives

German stellarator revamped to run longer, hotter, compete with tokamaks

Wendelstein 7-X’s complex geometry was a nightmare to build but, when fired up, worked from the start.

Tokamaks have dominated the search for fusion energy for decades. Just as ITER, the world's largest and most expensive tokamak, nears completion in southern France, a smaller, twistier testbed will start up in Germany.

If the 16-meter-wide stellarator can match or outperform similar-size tokamaks, fusion experts may rethink their future. Stellarators can keep their superhot gases stable enough to fuse nuclei and produce energy. They can theoretically run forever, but tokamaks must pause to reset their magnet coils.

The €1 billion German machine, Wendelstein 7-X (W7-X), is already getting "tokamak-like performance" in short runs, claims plasma physicist David Gates, preventing particles and heat from escaping the superhot gas. If W7-X can go long, "it will be ahead," he says. "Stellarators excel" Eindhoven University of Technology theorist Josefine Proll says, "Stellarators are back in the game." A few of startup companies, including one that Gates is leaving Princeton Plasma Physics Laboratory, are developing their own stellarators.

W7-X has been running at the Max Planck Institute for Plasma Physics (IPP) in Greifswald, Germany, since 2015, albeit only at low power and for brief runs. W7-X's developers took it down and replaced all inner walls and fittings with water-cooled equivalents, allowing for longer, hotter runs. The team reported at a W7-X board meeting last week that the revised plasma vessel has no leaks. It's expected to restart later this month to show if it can get plasma to fusion-igniting conditions.

Wendelstein 7-X’s twisting inner surface is now water cooled, enabling longer runs

Wendelstein 7-X's water-cooled inner surface allows for longer runs.


Both stellarators and tokamaks create magnetic gas cages hot enough to melt metal. Microwaves or particle beams heat. Extreme temperatures create a plasma, a seething mix of separated nuclei and electrons, and cause the nuclei to fuse, releasing energy. A fusion power plant would use deuterium and tritium, which react quickly. Non-energy-generating research machines like W7-X avoid tritium and use hydrogen or deuterium instead.

Tokamaks and stellarators use electromagnetic coils to create plasma-confining magnetic fields. A greater field near the hole causes plasma to drift to the reactor's wall.

Tokamaks control drift by circulating plasma around a ring. Streaming creates a magnetic field that twists and stabilizes ionized plasma. Stellarators employ magnetic coils to twist, not plasma. Once plasma physicists got powerful enough supercomputers, they could optimize stellarator magnets to improve plasma confinement.

W7-X is the first large, optimized stellarator with 50 6- ton superconducting coils. Its construction began in the mid-1990s and cost roughly twice the €550 million originally budgeted.

The wait hasn't disappointed researchers. W7-X director Thomas Klinger: "The machine operated immediately." "It's a friendly machine." It did everything we asked." Tokamaks are prone to "instabilities" (plasma bulging or wobbling) or strong "disruptions," sometimes associated to halted plasma flow. IPP theorist Sophia Henneberg believes stellarators don't employ plasma current, which "removes an entire branch" of instabilities.

In early stellarators, the magnetic field geometry drove slower particles to follow banana-shaped orbits until they collided with other particles and leaked energy. Gates believes W7-X's ability to suppress this effect implies its optimization works.

W7-X loses heat through different forms of turbulence, which push particles toward the wall. Theorists have only lately mastered simulating turbulence. W7-X's forthcoming campaign will test simulations and turbulence-fighting techniques.

A stellarator can run constantly, unlike a tokamak, which pulses. W7-X has run 100 seconds—long by tokamak standards—at low power. The device's uncooled microwave and particle heating systems only produced 11.5 megawatts. The update doubles heating power. High temperature, high plasma density, and extensive runs will test stellarators' fusion power potential. Klinger wants to heat ions to 50 million degrees Celsius for 100 seconds. That would make W7-X "a world-class machine," he argues. The team will push for 30 minutes. "We'll move step-by-step," he says.

W7-X's success has inspired VCs to finance entrepreneurs creating commercial stellarators. Startups must simplify magnet production.

Princeton Stellarators, created by Gates and colleagues this year, has $3 million to build a prototype reactor without W7-X's twisted magnet coils. Instead, it will use a mosaic of 1000 HTS square coils on the plasma vessel's outside. By adjusting each coil's magnetic field, operators can change the applied field's form. Gates: "It moves coil complexity to the control system." The company intends to construct a reactor that can fuse cheap, abundant deuterium to produce neutrons for radioisotopes. If successful, the company will build a reactor.

Renaissance Fusion, situated in Grenoble, France, raised €16 million and wants to coat plasma vessel segments in HTS. Using a laser, engineers will burn off superconductor tracks to carve magnet coils. They want to build a meter-long test segment in 2 years and a full prototype by 2027.

Type One Energy in Madison, Wisconsin, won DOE money to bend HTS cables for stellarator magnets. The business carved twisting grooves in metal with computer-controlled etching equipment to coil cables. David Anderson of the University of Wisconsin, Madison, claims advanced manufacturing technology enables the stellarator.

Anderson said W7-X's next phase will boost stellarator work. “Half-hour discharges are steady-state,” he says. “This is a big deal.”



1 year ago

Since I'm a scientist, I oppose biohacking

Understanding your own energy depletion and restoration is how to truly optimize

Photo: Towfiqu barbhuiya / Unsplash

Hack has meant many bad things for centuries. In the 1800s, a hack was a meager horse used to transport goods.

Modern usage describes a butcher or ax murderer's cleaver chop. The 1980s programming boom distinguished elegant code from "hacks". Both got you to your goal, but the latter made any programmer cringe and mutter about changing the code. From this emerged the hacker trope, the friendless anti-villain living in a murky hovel lit by the computer monitor, eating junk food and breaking into databases to highlight security system failures or steal hotdog money.

Remember the 1995 movie, Hackers, in which a bunch of super cool programmers (said no one ever) get caught up in a plot to destroy the world and only teenybopper Angelina Jolie and her punk rock gang of nerd-bots can use their lightening quick typing skills to save the world? Remember public phones?

Now, start-a-billion-dollar-business-from-your-garage types have shifted their sights from app development to DIY biology, coining the term "bio-hack". This is a required keyword and meta tag for every fitness-related podcast, book, conference, app, or device.

Bio-hacking involves bypassing your body and mind's security systems to achieve a goal. Many biohackers' initial goals were reasonable, like lowering blood pressure and weight. Encouraged by their own progress, self-determination, and seemingly exquisite control of their biology, they aimed to outsmart aging and death to live 180 to 1000 years (summarized well in this article).

With this grandiose north star, the hunt for novel supplements and genetic engineering began.

Companies selling do-it-yourself biological manipulations cite lab studies in mice as proof of their safety and success in reversing age-related diseases or promoting longevity in humans (the goal changes depending on whether a company is talking to the federal government or private donors).

The FDA is slower than science, they say. Why not alter your biochemistry by buying pills online, editing your DNA with a CRISPR kit, or using a sauna delivered to your home? How about a microchip or electrical stimulator?

What could go wrong?

I'm not the neo-police, making citizen's arrests every time someone introduces a new plumbing gadget or extrapolates from animal research on resveratrol or catechins that we should drink more red wine or eat more chocolate. As a scientist who's spent her career asking, "Can we get better?" I've come to view bio-hacking as misguided, profit-driven, and counterproductive to its followers' goals.

We're creatures of nature. Despite all the new gadgets and bio-hacks, we still use Roman plumbing technology, and the best way to stay fit, sharp, and happy is to follow a recipe passed down since the beginning of time. Bacteria, plants, and all natural beings are rhythmic, with alternating periods of high activity and dormancy, whether measured in seconds, hours, days, or seasons. Nature repeats successful patterns.

During the Upstate, every cell in your body is naturally primed and pumped full of glycogen and ATP (your cells' energy currencies), as well as cortisol, which supports your muscles, heart, metabolism, cognitive prowess, emotional regulation, and general "get 'er done" attitude. This big energy release depletes your batteries and requires the Downstate, when your subsystems recharge at the cellular level.

Downstates are when you give your heart a break from pumping nutrient-rich blood through your body; when you give your metabolism a break from inflammation, oxidative stress, and sympathetic arousal caused by eating fast food — or just eating too fast; or when you give your mind a chance to wander, think bigger thoughts, and come up with new creative solutions. When you're responding to notifications, emails, and fires, you can't relax.

Every biological plant and animal is regulated by rhythms of energy-depleting Upstate and energy-restoring Downstates.

Downstates aren't just for consistently recharging your battery. By spending time in the Downstate, your body and brain get extra energy and nutrients, allowing you to grow smarter, faster, stronger, and more self-regulated. This state supports half-marathon training, exam prep, and mediation. As we age, spending more time in the Downstate is key to mental and physical health, well-being, and longevity.

When you prioritize energy-demanding activities during Upstate periods and energy-replenishing activities during Downstate periods, all your subsystems, including cardiovascular, metabolic, muscular, cognitive, and emotional, hum along at their optimal settings. When you synchronize the Upstates and Downstates of these individual rhythms, their functioning improves. A hard workout causes autonomic stress, which triggers Downstate recovery.

This zig-zag trajectory of performance improvement illustrates that getting better at anything in life isn’t a straight shot. The close-up box shows how prioritizing Downstate recovery after an Upstate exertion (e.g., hard workout) leads to RECOVERYPLUS. Image from The Power of the Downstate by Sara C. Mednick PhD.

By choosing the right timing and type of exercise during the day, you can ensure a deeper recovery and greater readiness for the next workout by working with your natural rhythms and strengthening your autonomic and sleep Downstates.

Morning cardio workouts increase deep sleep compared to afternoon workouts. Timing and type of meals determine when your sleep hormone melatonin is released, ushering in sleep.

Rhythm isn't a hack. It's not a way to cheat the system or the boss. Nature has honed its optimization wisdom over trillions of days and nights. Stop looking for quick fixes. You're a whole system made of smaller subsystems that must work together to function well. No one pill or subsystem will make it all work. Understanding and coordinating your rhythms is free, easy, and only benefits you.

Dr. Sara C. Mednick is a cognitive neuroscientist at UC Irvine and author of The Power of the Downstate (HachetteGO)

Katherine Kornei

Katherine Kornei

2 years ago

The InSight lander from NASA has recorded the greatest tremor ever felt on Mars.

The magnitude 5 earthquake was responsible for the discharge of energy that was 10 times greater than the previous record holder.

Any Martians who happen to be reading this should quickly learn how to duck and cover.

NASA's Jet Propulsion Laboratory in Pasadena, California, reported that on May 4, the planet Mars was shaken by an earthquake of around magnitude 5, making it the greatest Marsquake ever detected to this point. The shaking persisted for more than six hours and unleashed more than ten times as much energy as the earthquake that had previously held the record for strongest.

The event was captured on record by the InSight lander, which is operated by the United States Space Agency and has been researching the innards of Mars ever since it touched down on the planet in 2018 (SN: 11/26/18). The epicenter of the earthquake was probably located in the vicinity of Cerberus Fossae, which is located more than 1,000 kilometers away from the lander.

The surface of Cerberus Fossae is notorious for being broken up and experiencing periodic rockfalls. According to geophysicist Philippe Lognonné, who is the lead investigator of the Seismic Experiment for Interior Structure, the seismometer that is onboard the InSight lander, it is reasonable to assume that the ground is moving in that area. "This is an old crater from a volcanic eruption."

Marsquakes, which are similar to earthquakes in that they give information about the interior structure of our planet, can be utilized to investigate what lies beneath the surface of Mars (SN: 7/22/21). And according to Lognonné, who works at the Institut de Physique du Globe in Paris, there is a great deal that can be gleaned from analyzing this massive earthquake. Because the quality of the signal is so high, we will be able to focus on the specifics.

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Alexander Nguyen

Alexander Nguyen

1 year ago

A Comparison of Amazon, Microsoft, and Google's Compensation

Learn or earn

In 2020, I started software engineering. My base wage has progressed as follows:

Amazon (2020): $112,000

Microsoft (2021): $123,000

Google (2022): $169,000

I didn't major in math, but those jumps appear more than a 7% wage increase. Here's a deeper look at the three.

The Three Categories of Compensation

Most software engineering compensation packages at IT organizations follow this format.

Minimum Salary

Base salary is pre-tax income. Most organizations give a base pay. This is paid biweekly, twice monthly, or monthly.

Recruiting Bonus

Sign-On incentives are one-time rewards to new hires. Companies need an incentive to switch. If you leave early, you must pay back the whole cost or a pro-rated amount.


Equity is complex and requires its own post. A company will promise to give you a certain amount of company stock but when you get it depends on your offer. 25% per year for 4 years, then it's gone.

If a company gives you $100,000 and distributes 25% every year for 4 years, expect $25,000 worth of company stock in your stock brokerage on your 1 year work anniversary.

Performance Bonus

Tech offers may include yearly performance bonuses. Depends on performance and funding. I've only seen 0-20%.

Engineers' overall compensation usually includes:

Base Salary + Sign-On + (Total Equity)/4 + Average Performance Bonus

Amazon: (TC: 150k)

Photo by ANIRUDH on Unsplash

Base Pay System

Amazon pays Seattle employees monthly on the first work day. I'd rather have my money sooner than later, even if it saves processing and pay statements.

The company upped its base pay cap from $160,000 to $350,000 to compete with other tech companies.

Performance Bonus

Amazon has no performance bonus, so you can work as little or as much as you like and get paid the same. Amazon is savvy to avoid promising benefits it can't deliver.

Sign-On Bonus

Amazon gives two two-year sign-up bonuses. First-year workers could receive $20,000 and second-year workers $15,000. It's probably to make up for the company's strange equity structure.

If you leave during the first year, you'll owe the entire money and a prorated amount for the second year bonus.


Most organizations prefer a 25%, 25%, 25%, 25% equity structure. Amazon takes a different approach with end-heavy equity:

  • the first year, 5%

  • 15% after one year.

  • 20% then every six months

We thought it was constructed this way to keep staff longer.

Microsoft (TC: 185k)

Photo by Louis-Philippe Poitras on Unsplash

Base Pay System

Microsoft paid biweekly.

Gainful Performance

My offer letter suggested a 0%-20% performance bonus. Everyone will be satisfied with a 10% raise at year's end.

But misleading press where the budget for the bonus is doubled can upset some employees because they won't earn double their expected bonus. Still barely 10% for 2022 average.

Sign-On Bonus

Microsoft's sign-on bonus is a one-time payout. The contract can require 2-year employment. You must negotiate 1 year. It's pro-rated, so that's fair.


Microsoft is one of those companies that has standard 25% equity structure. Except if you’re a new graduate.

In that case it’ll be

  • 25% six months later

  • 25% each year following that

New grads will acquire equity in 3.5 years, not 4. I'm guessing it's to keep new grads around longer.

Google (TC: 300k)

Photo by Rubaitul Azad on Unsplash

Base Pay Structure

Google pays biweekly.

Performance Bonus

Google's offer letter specifies a 15% bonus. It's wonderful there's no cap, but I might still get 0%. A little more than Microsoft’s 10% and a lot more than Amazon’s 0%.

Sign-On Bonus

Google gave a 1-year sign-up incentive. If the contract is only 1 year, I can move without any extra obligations.

Not as fantastic as Amazon's sign-up bonuses, but the remainder of the package might compensate.


We covered Amazon's tail-heavy compensation structure, so Google's front-heavy equity structure may surprise you.

Annual structure breakdown

  • 33% Year 1

  • 33% Year 2

  • 22% Year 3

  • 12% Year 4

The goal is to get them to Google and keep them there.

Final Thoughts

This post hopefully helped you understand the 3 firms' compensation arrangements.

There's always more to discuss, such as refreshers, 401k benefits, and business discounts, but I hope this shows a distinction between these 3 firms.

James Brockbank

1 year ago

Canonical URLs for Beginners

Canonicalization and canonical URLs are essential for SEO, and improper implementation can negatively impact your site's performance.

Canonical tags were introduced in 2009 to help webmasters with duplicate or similar content on multiple URLs.

To use canonical tags properly, you must understand their purpose, operation, and implementation.

Canonical URLs and Tags

Canonical tags tell search engines that a certain URL is a page's master copy. They specify a page's canonical URL. Webmasters can avoid duplicate content by linking to the "canonical" or "preferred" version of a page.

How are canonical tags and URLs different? Can these be specified differently?


Canonical tags are found in an HTML page's head></head> section.

<link rel="canonical" href="" />

These can be self-referencing or reference another page's URL to consolidate signals.

Canonical tags and URLs are often used interchangeably, which is incorrect.

The rel="canonical" tag is the most common way to set canonical URLs, but it's not the only way.

Canonical URLs

What's a canonical link? Canonical link is the'master' URL for duplicate pages.

In Google's own words:

A canonical URL is the page Google thinks is most representative of duplicate pages on your site.

— Google Search Console Help

You can indicate your preferred canonical URL. For various reasons, Google may choose a different page than you.

When set correctly, the canonical URL is usually your specified URL.

Canonical URLs determine which page will be shown in search results (unless a duplicate is explicitly better for a user, like a mobile version).

Canonical URLs can be on different domains.

Other ways to specify canonical URLs

Canonical tags are the most common way to specify a canonical URL.

You can also set canonicals by:

  • Setting the HTTP header rel=canonical.

  • All pages listed in a sitemap are suggested as canonicals, but Google decides which pages are duplicates.

  • Redirects 301.

Google recommends these methods, but they aren't all appropriate for every situation, as we'll see below. Each has its own recommended uses.

Setting canonical URLs isn't required; if you don't, Google will use other signals to determine the best page version.

To control how your site appears in search engines and to avoid duplicate content issues, you should use canonicalization effectively.

Why Duplicate Content Exists

Before we discuss why you should use canonical URLs and how to specify them in popular CMSs, we must first explain why duplicate content exists. Nobody intentionally duplicates website content.

Content management systems create multiple URLs when you launch a page, have indexable versions of your site, or use dynamic URLs.

Assume the following URLs display the same content to a user:









A search engine sees eight duplicate pages, not one.

  • URLs #1 and #2: the CMS saves product URLs with and without the category name.

  • #3, #4, and #5 result from the site being accessible via HTTP, HTTPS, www, and non-www.

  • #6 is a subdomain mobile-friendly URL.

  • URL #7 lacks URL #2's trailing slash.

  • URL #8 uses a capital "A" instead of a lowercase one.

Duplicate content may also exist in URLs like:

Duplicate content is easy to create.

Canonical URLs help search engines identify different page variations as a single URL on many sites.

SEO Canonical URLs

Canonical URLs help you manage duplicate content that could affect site performance.

Canonical URLs are a technical SEO focus area for many reasons.

Specify URL for search results

When you set a canonical URL, you tell Google which page version to display.

Which would you click?

First, probably.

Canonicals tell search engines which URL to rank.

Consolidate link signals on similar pages

When you have duplicate or nearly identical pages on your site, the URLs may get external links.

Canonical URLs consolidate multiple pages' link signals into a single URL.

This helps your site rank because signals from multiple URLs are consolidated into one.

Syndication management

Content is often syndicated to reach new audiences.

Canonical URLs consolidate ranking signals to prevent duplicate pages from ranking and ensure the original content ranks.

Avoid Googlebot duplicate page crawling

Canonical URLs ensure that Googlebot crawls your new pages rather than duplicated versions of the same one across mobile and desktop versions, for example.

Crawl budgets aren't an issue for most sites unless they have 100,000+ pages.

How to Correctly Implement the rel=canonical Tag

Using the header tag rel="canonical" is the most common way to specify canonical URLs.

Adding tags and HTML code may seem daunting if you're not a developer, but most CMS platforms allow canonicals out-of-the-box.

These URLs each have one product.

How to Correctly Implement a rel="canonical" HTTP Header

A rel="canonical" HTTP header can replace canonical tags.

This is how to implement a canonical URL for PDFs or non-HTML documents.

You can specify a canonical URL in your site's.htaccess file using the code below.

<Files "file-to-canonicalize.pdf"> Header add Link "<>; rel=\"canonical\"" </Files>

301 redirects for canonical URLs

Google says 301 redirects can specify canonical URLs.

Only the canonical URL will exist if you use 301 redirects. This will redirect duplicates.

This is the best way to fix duplicate content across:

  • HTTPS and HTTP

  • Non-WWW and WWW

  • Trailing-Slash and Non-Trailing Slash URLs

On a single page, you should use canonical tags unless you can confidently delete and redirect the page.

Sitemaps' canonical URLs

Google assumes sitemap URLs are canonical, so don't include non-canonical URLs.

This does not guarantee canonical URLs, but is a best practice for sitemaps.

Best-practice Canonical Tag

Once you understand a few simple best practices for canonical tags, spotting and cleaning up duplicate content becomes much easier.

Always include:

One canonical URL per page

If you specify multiple canonical URLs per page, they will likely be ignored.

Correct Domain Protocol

If your site uses HTTPS, use this as the canonical URL. It's easy to reference the wrong protocol, so check for it to catch it early.

Trailing slash or non-trailing slash URLs

Be sure to include trailing slashes in your canonical URL if your site uses them.

Specify URLs other than WWW

Search engines see non-WWW and WWW URLs as duplicate pages, so use the correct one.

Absolute URLs

To ensure proper interpretation, canonical tags should use absolute URLs.

So use:

<link rel="canonical" href="" />

And not:

<link rel="canonical" href="/page-a/" />

If not canonicalizing, use self-referential canonical URLs.

When a page isn't canonicalizing to another URL, use self-referencing canonical URLs.

Canonical tags refer to themselves here.

Common Canonical Tags Mistakes

Here are some common canonical tag mistakes.

301 Canonicalization

Set the canonical URL as the redirect target, not a redirected URL.

Incorrect Domain Canonicalization

If your site uses HTTPS, don't set canonical URLs to HTTP.

Irrelevant Canonicalization

Canonicalize URLs to duplicate or near-identical content only.

SEOs sometimes try to pass link signals via canonical tags from unrelated content to increase rank. This isn't how canonicalization should be used and should be avoided.

Multiple Canonical URLs

Only use one canonical tag or URL per page; otherwise, they may all be ignored.

When overriding defaults in some CMSs, you may accidentally include two canonical tags in your page's <head>.

Pagination vs. Canonicalization

Incorrect pagination can cause duplicate content. Canonicalizing URLs to the first page isn't always the best solution.

Canonicalize to a 'view all' page.

How to Audit Canonical Tags (and Fix Issues)

Audit your site's canonical tags to find canonicalization issues.

SEMrush Site Audit can help. You'll find canonical tag checks in your website's site audit report.

Let's examine these issues and their solutions.

No Canonical Tag on AMP

Site Audit will flag AMP pages without canonical tags.

Canonicalization between AMP and non-AMP pages is important.

Add a rel="canonical" tag to each AMP page's head>.

No HTTPS redirect or canonical from HTTP homepage

Duplicate content issues will be flagged in the Site Audit if your site is accessible via HTTPS and HTTP.

You can fix this by 301 redirecting or adding a canonical tag to HTTP pages that references HTTPS.

Broken canonical links

Broken canonical links won't be considered canonical URLs.

This error could mean your canonical links point to non-existent pages, complicating crawling and indexing.

Update broken canonical links to the correct URLs.

Multiple canonical URLs

This error occurs when a page has multiple canonical URLs.

Remove duplicate tags and leave one.

Canonicalization is a key SEO concept, and using it incorrectly can hurt your site's performance.

Once you understand how it works, what it does, and how to find and fix issues, you can use it effectively to remove duplicate content from your site.

Canonicalization SEO Myths



2 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 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:
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