My non-recommendation comes down to not that this is necessarily a bad product, but by most players standards, The Polynomial is hardly a game.
Load up the game and you're brought up to the first loaded game world and a menu to configure a variety of options. The "world" you are in immediately shows you a list of "what items does what" much like an old arcade game or a flash game, though, I had to load into the game and aim my camera around to read the entire menu. The game worlds have various fractal based visualizations which appear to be various streams of light created by colorful small orbs.
Before I go into the details of the game, while there are definitely available combat game mechanics, it feels more like a 3D Space Flight simulator edition of the visualizations you would see in something like WinAMP. The game is damn pretty and it is interesting for a little while to see how the steams and balls of light react to the music (including your own injected music). You fly around these visualizations much like a 6 Degrees of Freedom (6DOF) shooter like Descent or Retrovirus. Ghosts around the level can speed you up and heal some of your damage, while pac-man like chompers try to eat said ghosts and also shoot to kill you. You can also shoot in the same fashion in the form of a endless steam of blue orbs. It's week, but its infinite and perfectly accurrate with some bullet travel, so you need to lead targets sometimes. That appears to be the only way to harm these enemies. There are other items that protect you from damage, give you power ups like more speed and damage, and let you travel to other worlds and visulazations. The only true goal it seems to to see how high a score you can get in certain allotted time frames to be compared on the leaderboard. Strangely, the first time I played the game, I didn't have a HUD to show my kills, saves, points and etc on my first play, but on my second loading of the game, i saw a big list in the upper left hand corner of the screen - maybe I changed an option in error. Finally, you can also make and share your own levels by choosing from various Seeds a la Minecraft and messing with a multitude of visualizations and various options. The music you choose is in a separate menu for all players depending on what they imported, so I don't think it matters what music you "choose" for a level - the game is dependent on the music the player has (which there is some of in the game).
If you're expecting gameplay experience like Beat Hazard or Symphony, then you probably won't be entertained as I found the game the be quite boring and confusing and it felt like it was missing something. It feels as the game was meant to be a "interactive visulization" and the developer felt they had to add game mechanics to make it "playable". That said, it appears to be a potentially interesting way to "fly through your music" - I may give it another chance in easy modes just to "chillax" and soak in the visual experience, but I generally rather "play" my games. Though thanks to the indie scene, there are much more entertaining and rewarding "import your own music titles" from a pure gameplay perspective. However, as made obvious by the screenshots and video, the game is gorgeous.
This is probably best seen as a love letter to the fractal/math/demoscene/visualization community, so people of those types should potentially enjoy this. This game is a tough recommendation for core gamers and music lovers. I generally watch TotalBiscuits WTF videos of a game if I can find it, though for whatever reason I didn't look for one this time (and somewhat regret that). However, TB has made a video of this to give you an idea of what is going on.